I've looked in the DMG and the PHB, but have only found ancillary uses of the shield. The Shield Master feat (PHB, p.170) gives you neat perks, but doesn't act as a weapon (beyond the bonus shove action).
Does shield proficiency automatically grant you shield bashing?
What would reasonable damage be?
And do the old rules apply for losing the defense bonus when using it as a weapon?
Does it count as a light weapon? Can I use it as a two-weapon bonus action?
A general preface: each edition of D&D is its own game, and rules should be evaluated as such. Just because things happened in old rule sets does not mean that it will in 5e. Though it can be helpful to look at old editions for inspiration, you should be careful of the rules environs of the edition you are trying to modify when making house rules.
First, Shield Bash is already a named mechanic in 5E: it's the additional power associated with Shield Master that allows you to attempt to push someone as a bonus action when you make an attack. Also, you can do this same thing as a Full Action (or a piece of an attack action) using the contest rules and the "Shove" improvised action (Players Basic 74). That said, that's not what you're asking about here. What you want to know is how to do damage with your shield.
If you want to hit someone with your shield, it is treated as an improvised weapon attack.
The question then becomes does it resemble any existing weapon we have game statistics for? as per the "Improvised Weapons" section on pg 47 of Player's Basic. Looking at the list, I don't see an obvious resemblance (though feel free to make your own observations/additions). Thus it acts as a 1d4 weapon and proficiency is only granted if you have proficiency in improvised weapons.
There is also no provision for removing the defense bonus. In fact Shield Master improves the shield with attacks without removing the defense bonus so there seems no need to import that rule from a prior edition. (Additionally, it seems to be punitive beyond the spirit of 5e's rules.)
So, ultimately, shield bashing is a flavorful power that is probably not a good mechanical choice. If you want to do it, talk to your DM, but without house rules, it's not a great use of an action. If you want the mechanics, and don't mind not doing damage on the push, Shield Master's push is probably the best you're going to do.
This is adjunct to Wax Eagle answer.
Historically Shield Bashing was used part of the whole package of skills learned when using a sword and shield. This included maneuvers what we could consider to be a type of grappling along with full body contact. What they amounted to in an abstract was to get the opponent prone or out of position so you had an tactical advantage
In D&D 5e with its level of abstraction, the main use of a shield bash would be to knock the target prone thus granting advantage to the attack on a follow up attack. This doesn't work really well for a low level character as generally they only have the ability to execute one attack action in a round.
To knock down a character to prone, you can use a attack option to shove your opponent. If you win the ensuing Str (Athletic) version Str (Athletic)/Dex (Acrobatic) check the opponent is knocked prone. This is found on page 195
However this can be done even if you wielding only a sword single handed. It meant to abstract the whole body contact/grappling part of fighting with melee weapons.
The shove require an attack so you need to be able to generate multiple attacks in a round to take advantage of it. Otherwise the next round the opponent will just use half of his move and get up.
You and your referee may want to consider as a house rule to allow the use of a shield execute a shove attack as a bonus action. Basically treat it as a character with two weapons. If this is felt to be too generous you can still allow this but say you can't add your proficiency modifier to your strength roll. This is similar to the two weapon rule where you can't add your damage modifier (unless it is negative) to the second roll.
My character uses this technique, and I believe we are doing it both RAW and RAI. So here we go:
To sum it up:
If you do all this, then you get:
Considering the investment (one feat, one background feature, one fighting style), I think it is well balanced.
I might add the Shield Master feat to the mix for my next feat. Apart from the fact that I only have one bonus action per round, the benefits all stack.
With the rules as written, shields are not weapons. They are not considered capable of doing enough damage to warrant a ruling. Their primary purpose is to grant +2AC, and their primary drawback is that you can't use that hand for another purpose such as holding a weapon.
This becomes clear when you look at Shield Master Feat (PHB 170), where we might reasonably have expected to see damaging shield bash rules. It allows you to attempt to push someone as a bonus action when you make an Attack Action. If you hit, you knock the target prone or 5 feet away from you. Note that it doesn't do any damage.
As an aside, the "Shove" Attack Action can achieve the same result (PHB p195); it is available to all, though harder to pull off. Contest Strength (Athletics) vs the the target's. It uses one of your attacks if you have more than one. Success knocks the target prone or 5 feet away from you. But note: Shove makes no mention of using a shield.
The shield can only be used to cause damage if it is used as an Improvised Weapon. Importantly, while doing so, it is not functioning as a shield.
When using a shield as an improvised weapon (PHB p147-148) , you are Two-Weapon Fighting (PHB p195). Because of this, you get a bonus action with which to attack with the shield. Also, don't add your ability modifier to the damage of that bonus attack unless the modifier is negative.
The shield bears no similarity to an existing weapon so does 1d4 damage with no chance to add a proficiency bonus to your hit roll unless you have the Tavern Brawler feat.
You would need to invest in the Dual Wielder Feat (PHB p165) to use it since it is not a light weapon. You lose the +2AC from the shield as it ceases to act as a shield, but gain +1AC for using two 'weapons'.
To attack with a shield in your off-hand is an improvised weapon bonus attack requiring the Feat Dual Wielder, doing d4 (minus any negative Strength modifier) damage, and reducing AC from +2 for a shield to +1 for Dual Wielder.
Characters with the Tavern Brawler feat roll and add their proficiency bonus.
In summary, you would only resort to it if you were disarmed of your main weapon.
Arguably you could then switch the shield from an off-hand improvised weapon to your main (improvised) weapon, allowing you to add your Strength Bonus to your damage, and no longer requiring Dual Wielder Feat. In this case...
To attack with a shield in your weapon-hand is an improvised weapon attack, doing d4 (plus your Strength modifier) damage. Reduce AC by +2 while your shield is being wielded as a weapon.
Characters with the Tavern Brawler feat roll and add their proficiency bonus.
Once you have dual wielding, you're better off using a weapon in that hand to do d8 damage.
The Shield Master feat is a better option for shield bashing. As a
bonus action you can shove which is way better than 1d4 damage most
of the time, since you give all your regular attacks advantage and
some of your allies, after you knock the target prone.
That a shield bash only pushes an opponent back tally's with real world usage. In medieval combat, a shield bash was used to wrong-foot or floor an opponent. It was the weapon strike afterwards that did the damage. Trying to do actual damage with your shield was likely to leave you defenceless. It makes a poor weapon against real armour. Better and safer to use your actual weapon for the purpose for which it was designed.
The popular idea of the devastating shield bash is perpetuated by fiction, where a warrior's legendary strength is often expressed visually by his ability to literally bat minions away with his shield.
During a reenactment exercise, I was hit with a shield using the flat of it, the sharp edge and the point at the bottom. The shield was accurate in size and weight. It hurt, but not as much as a blow from an axe or sword would, with its leverage and momentum (thankfully, these were less authentic).
I've also had the privilege of being able to talk to a stunt man who specialises in horse and sword fighting (one of the Devils Horsemen team). He said, "I might push someone back so I can swing with the sword. If they are on top of my shield they deserve it. Swinging out with the shield... I wouldn't, especially if there are a lot of people around." It would create an opportunity for an enemy.
A lot of players want it to be true than you can deliver extra damage during your round with the shield, but this has wide ranging implications for class balance and fairness. It also owns up arguments about whether a head but with a helmet, or a punch with a mailed fist should do more damage than in the RAW.
Jemmy Crawford, a cowriter of 5e said in a tweet that when used as an Improvised weapon, a shield retains its characteristics as a shield, therefore continuing to provide +2AC. Sacrilegious though it may be, I have to say I think he's made a mistake by coming out with that. His case was that a spell focus item is still a spell Focus item even when it is being used for improvised attacks, so the same should be true of a shield and it's armour class bonus, but I would argue that a shield is no longer in a position to confirm defensive qualities if it is being waved around as a weapon. Drawing this distinction also helps keep things clearer. The Shield is considered as an improvised weapon and abides by those rules without the muddiness of adding rules that are not in the book.
In researching this, I think the rules are sound (if hard to piece together) and do not have a glaring gap in them, as some people argue.
As for damage, I see someone quoted the gladiator, but the MM has lizardfolk using a "spiked shield" that does 1d6+Str damage, which can be used as a weapon with which they can make one of their two melee attacks as part of their Multiattack action (MM, p. 204):
Multiattack. The lizardfolk makes two melee attacks, each one with a different weapon.
[...]
Spiked Shield. Melee Weapon Attack: +4 to hit, reach 5 ft. one target. Hit: 5 (1d6 + 2) piercing damage.
I would still be inclined to have a regular shield do 1d4 (+ Str mod) damage.
There are no rules for specifically attacking with a shield.
There are two options you could use: improvised weapon and unarmed attack.
You could say that a shield is an improvised weapon doing, say, 1d4+STR bludgeoning damage.
You would then have to rule on whether it is a light weapon or a finesse weapon, and so on, to answer questions like "does a rogue get sneak attack damage when attacking with a shield?" and "does two-weapon fighting apply to attacking with a shield?" and "can an Eldritch Knight bond with a shield?".
You should also think about helms and gauntlets and boots and so on. If a shield is an improvised weapon then what about a helm (you can head butt a foe with it)? Does this mean an Eldritch Knight can weapon-bond with their magical helm?
You could say a shield bash is covered by the unarmed rules (my personal ruling).
This separates things you pick up and drop (improvised weapons) from things you wear (gauntlets, boots, helm, shield). This means that attacking someone with a shield is mechanically the same as attacking them with a helm (head butt) or armour (elbow strike, knee strike) or gauntlets (punch) or boots (kick). It means you can say "no" straight away when the player of an eldritch knight asks to weapon-bond with their magical gauntlets that they use to punch people.
I rule that unarmed attacks do 1+STR damage for no armour and light armour, 2+STR damage for medium armour and/or shield, 3+STR for heavy armour. I figure that a headbutt or elbow strike while wearing plate should hurt more.
Unarmed strikes are not weapons, so they don't trigger some feats and features.
The game does have a rule called "shield bash". It is not an attack, however. It is a shove granted by the Shield Mastery feat.
There is another precedent in the game for a shield bash, not in the PHB but in the MM - in the gladiator (p. 346) NPC statblock, as one of its actions:
Shield Bash. Melee Weapon Attack: +7 to hit, reach 5 ft., one creature. Hit: 9 (2d4 + 4) bludgeoning damage. If the target is a Medium or smaller creature, it must succeed on a DC 15 Strength saving throw or be knocked prone.
Given the the Str 18(+4) and the damage above, it appears that there is precedence for a shield to have damage 2d4 + Str and be a Melee Weapon Attack.
This type of "monster" however also appears to bend rules and include a variation on the shield master shove built into the basic attack.
Bottom line however is that it looks like as a weapon a shield does 2d4 + Str, and if proficient with shields I see no reason why it can't be used as a weapon in this way.
The next big question however is, if it is a +1 shield does that include +1 as a weapon attack/damage?
As of 5/1/2015 there is no official WOTC answer. I like allowing it to be used as an improvised weapon (Proficiency granted with any shield proficiency) with 1d4+str damage. AC bonus (as well as any other effects such as the "evasion" granted by Shield Master) from shield is forfeited until next turn.
I would rule that a shield bash should be treated like an offhand attack under the two weapon fighting feature. Since it is not a light weapon, I would count it as an improvised weapon that will do 1d4 bludgeoning damage with no positive Str bonus, and you lose the shield bonus to A.C. until the next turn.
