Many classes/ builds allow one to fulfill multiple party roles. For example, a Paladin is a hybrid leader/defender. What classes/builds are a good way to handle striker/controller? I'm looking for a class/build that will allow a character to attack enemies, dealing damage, and applying debuff/status effects.
Edited for heroes of shadow and up to Dragon 404.
Some definitions of terms:
From a general theory point of view, it is the paragon path that changes the role. A combined striker/controller is in many ways a contradiction in terms: a striker wants to apply the best debuff in the game as fast as possible (0 HP) whereas a controller wants to perform hard and soft control at the expense of damage (as a means of damage-mitigation or damage-augmentation).
Now, the tricky part is that these technical definitions do not necessarily correspond with the role definitions. While in heroic, most builds will indeed play according to their generic role, the PP and most paragon tier feats allow shifting of role. Despite this, any given build will have a primary intent, and if built right, a useful secondary intent. Just as paladins are off-leaders, so too can there be off-strikers or off-controllers.
In many ways, high damage is the easiest to achieve by observing two precepts: static-mod is king, multi-attacks are queen. Off-striker damage means that each target will generally be receiving /4 damage over the course of a round (i.e. significantly non-trivial).
Controller effects accumulate through three sources: feats, powers, class features. In short, there are two strategies: enhance a power that will be the character's centrepiece, or choose linked encounter powers to provide the intended effect. Dailies shall never be selected for damage (that's the pure-striker's job), instead they need to be selected for how they enhance the strategy of the character.
The rule of thumb is: it's easier to increase damage then it is to increase control.
Here, characters are lucky if they can get a status effect with their attacks. Controllers should be focused on minion popping and setting enemies up for CA or minor debuffs.
In all of these cases the pattern is: damage+effect.
The Thief with "Unbalancing trick" is an excellent striker/controller, as she can prone at-will, with remarkable damage.
The Hunter provides excellent control matched with the remarkably large damage dice of bows and crossbows. Rapid Shot is a minion killer, and Clever shot provides very solid control options.
The thief wants damage through Surprising Stab, the Hunter won't say no to weapon focus.
The bladesinger (Neverwinter) is a sword&spell controller. Their MBAs are bog-standard, but because they can slide, slow, prone, etc.. at will after an MBA from their "bladespells" they cna prove surprisingly effective. With feral armor (hide, level 7) they can even provide multi-target control with what (if you blur the image a little) appears to be a striker's nova-round.
The ardent and psion are excellent controller/strikers with demoralizing strike (it'll keep your entire career) for the ardent and dishearten and mind thrust for the psion. Both of these are in a way of soft control, debuffing defense or offense. The ardent actually can have better multi-target control dailies than the psion, due to the sheer amount of zone options she has. The psion's living missle, on the other hand, can shape the course of an encounter and do respectable damage.
The prevalence of psionic options isn't an accident. Due to their weaker class features and stronger at-wills, any given psionic class can be closely tailored to requirements, so long as there is sufficient support from dailies.
Monk provides excellent forced movement and AoE damage, though will be a bit weak on the debuffs.
Accuracy is the name of the game here, featwise.
For all divine classes, the "Power of the X" feats can offer a side benefit of control or damage, though the damage will be expressed more thoroughly later in paragon and epic.
The invoker with her hand of radiance and power of the moon can debuff 3 targets's reflex. Which is awesome in the right party. With mark of storm, she can shift each target one. And with 3 attacks, her contribution to party DPR through her static mod will be very very impressive. At level 1, divine bolts are a trap: enhancing them comes later. The invoker has excellent daily support for specific controller flavours, with some quite damaging options.
The mage has a hard time providing both damage and control, as her options tend to favour one over the other. Despite this, there are a significant number of excellent choices.
Accuracy feats are quite handy here.
Heroes of shadow introduced the executioner assassin. While normally an interesting if not completely stellar striker, Dragon 404 introduced the "Ninja" subclass. The ninja is an exceptional controller/striker: they have a 3 target minion-popping attack (that, with the right effort can turn into a very respectable source of damage on its own (ask a new question for details on that research)) and a proning at-will that provides exceptional soft control due to the shift 1 that it offers after prone. This attack can completely negate an enemy's turn by preventing them from standing and charging. The assassin poisons, especially delivered by the triple-target poisonous shuriken, become an excellent way to apply the status effects of slowed and dazed.
The seeker is a trap. She just doesn't have enough support. Beware of the huge number of choices the wizard gets, it's easy to become distracted. Warlocks and most striker classes will be focusing on damage these first few levels. Warlocks, specifically, are spectacularly easy to mis-build, and a controller focus tends to require looking through many resources.
The Binder, unfortunately, is also a trap. While a controller warlock is nice in theory, the powers are sufficiently lackluster to negate most "controllery" benefits while offering no striker-damage to compensate. If you'd like to play one, apply the changes suggested by square fireballs here.
The Vampire isn't even a trap. While they get forced movement at level 1, their single-target low-damage powers coupled with no meaningful choices suggests that they should be considered only if the player wants to play a vampire, not if they want to play an effective striker or controller. However, they can become far more interesting with hybridization or multiclassing.
In the middle of heroic, classes start gaining options with the option of an MC under their belt along with their enhancing feats. The joy of the fighter and seeker MC will come in paragon, but at 6, most classes will be focused on exploring their "core competencies"
Divine characters will want to focus on getting the most mileage out of their at-will. MBA characters will want to focus on damage. Psionic characters have a straightforward upgrade path at this point.
As a special note, Battleminds at level 7 start getting some fantastic striker/control options. Forceful reversal combines out of turn damage (multi-attacks!) with pushes (Forced movement!)
Arcane characters may want to dabble in some of the white-lotus training for debuffs, movement, or hindering terrain. Err on the side of later levels, however. Improve damage for now.
The thief should grab herself a spiked chain, in anticipation of paragon.
Druids are viable here, with Claw Gloves and feats to enhance their beast-form MBA.
Blackguards with access to some of the paladin dailies can become effective off-controllers, providing melee control through superior firepower and slowing effects.
First: everyone who wants to do damage takes lasting frost (and wintertouched if they need CA). It's boring, but it provides excellent damage.
The paragon path is where control or striker is really achieved. The best "mixandmatch" PP for role-crossing is Traveler's Harlequin, which allows you to take multiple MC feats along with a delicious cherrypicked level 20. The Level 11 is excellent battlefield rearrangement as well.
Ardent takes Stygian Adept, multiclassed to fighter for hindering shield with a staggering weapon. Then every melee attack with debilitating strike slows and slides, and amortized over the HP of all targeted mobs, does striker level damage.
Psions go with a Tiefling Firestarter.
Arena champion is a solid Rogue single-target control, the arena-fighting requirement lessens the attractiveness for thief.
Strong-arm enforcer is a solid choice for thief and rogue, with disheartening ambush providing the rattling.
Thieves really should go Harlequin though, as that opens up fighter MC on top of their spiked chain MC, which allows them to get the various fighter debuffing on MBA feats. They should also grab lashing flail (for slide) and deft blade (for attacking reflex, therefore acccuracy). That gets them slide 1 and prone, which is an absolutely vicious single-target control, as well as options for fairly trivially getting threatening reach.
Arcane casters suddenly wake up with the white lotus paragon feats, annoying enemies to no-end with difficult terrain around all allies. Warlocks and Mages should therefore look at Academy Master, Feytouched for Warlock Puppeteers, or Nightmare Weavers. Warlocks also love Stygian Adept if they have Dire Radiance and are specced for fear.
Mages or half-elves with illusion at-wills (read:Winged Horde) should absolutely take battleweaver. The trivial damage and AoE optimization of Winged Horde combined with a party-friendly slow (and at this level psychic lock) make it an source of control and damage par excellance. For more details, ask a question about optimizing this specific combo.
Hunters are queens in paragon, due to half-elf versatility.
There are many other half-elf options depending on the amount of damage you want to dial. Hand of Radiance is a valuable addition to almost any character in paragon, offering multi-target damage (and all the rest) with a remarkably small investment. With that said, it's not worth dabbling in half-elf cheese: either commit to the dilliante power being the centrepiece of the build, or don't commit feats to it.
A further half-elf or hybrid option is offered through the archery master feat (suggested here). By taking an RBA focused class and getting rapid shot, you can effectively turn that RBA into something quite exceptional. Special attention should be paid to the once-trapped seekers due to this option.
Sorcerers with Lightning Fury gain some impressive battlefield control.
By this point, most strikers can spec for some single-target control and most controllers can spec for damage, based on their choice of multi-class feats and PP.
In Epic, the choices open even further, mainly due to some exceptional epic feats for the radiant mafia (Invokers with Hand of radiance and an expanded crit range win here), with hunters still being exceptionally strong, ardents being great controllers (who can dish out damage based on their zones and close bursts), thieves being fantastic due to some trivial ways of getting 2 threatening reaches a day (warden MC through harlequin or barbarian MC, combined with the spiked chain feat series. Warlocks are also solid, though difficult to build to the same damage numbers, and so on and so forth.
Most classes can do single-target control by epic, simply by MCing appropriately, and the build-space is so large that it's difficult to make any specific statements, save that building a character should never be done without the handbooks.
The Warlock is built to fulfill that role. Most of its abilities allow you to push, pull, or teleport the enemy, and in the last case, to teleport yourself as well. As a complement, the star pact focuses on manipulating opponents' rolls in your favor or to the detriment of one's enemies.
You also get a fair number of status effects with abilities that can immobilize, weaken or impose a vulnerability.
A two-handed ranger built around the Net Training and World Serpent's Grasp feats can carry a net and trident for his Twin Strike weapons. Hit with the net, target is slowed; follow up with the spear and the target is prone. Choose powers that include a shift, and add the Swift Footwork feat, and now you can shift 3, 4, or sometimes 5 squares between strikes. Your enemies are slowed and/or prone, so they're going to be wasting their move actions getting back up on their feet; you'll be using yours moving to the next target.
The Essentials Wizard build, known as the Mage, does this rather well. They get at-wills that do a surprising amount of moving the battlefield or apply other status effects, always have Magic Missile for some reliable damage, and have some decently damaging Encounter/Daily powers.
Monk is another good one, the Iron Soul build does a lot of tactical positioning such that enemies are left unable to do certain things on virtually every round.
As Jadasc mentioned, Warlock is ok, but they are lacking in terms of DPR; don't expect to be putting out the same damage a Rogue does.
Seeker is another one, iirc. I haven't looked at them in depth but I seem to recall seeing a lot of status effects on their powers.