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Shillelagh imbues a quarterstaff with magic for a minute. This allows a caster to attack with the quarterstaff using his casting ability.
Polearm Master allows a player who attacks with a quarterstaff to get a second attack with the haft as bonus action.
Can I use my casting ability for the quarterstaff haft bonus attack?
I think so, but I am not sure. A good answer should include rule text and discussion to show why this is possible or why not.
The Shillelagh spell description, in relevant part:
For the duration, you can use your spellcasting ability instead of Strength for the attack and damage rolls of melee attacks using that weapon, and the weapon's damage die becomes a d8. (PHB, p. 275)
As written, the spell applies to all melee attacks with the affected weapon. It is not limited to only the Attack action. The Polearm Master feat, in relevant part:
When you take the Attack action and attack with only a [...] quarterstaff, you can use a bonus action to make a melee attack with the opposite end of the weapon. This attack uses the same ability modifier as the primary attack. The weapon’s damage die for this attack is a d4, and it deals bludgeoning damage. [Emphasis added] (PHB, p. 168)
The bonus attack is a melee attack, so you would be allowed to use your casting ability for the attack and damage rolls of the bonus attack (and the Attack action). However, while the Attack action uses a d8 for damage per the Shillelagh spell, the bonus attack uses a d4 damage die per the Polearm Master feat. This interaction was clarified by 5e designer and "official rules expert" Jeremy Crawford on 23 August 2014 in the following tweet exchange. When asked:
@JeremyECrawford If I cast Shillelagh on my quarterstaff and have Polearm Master feat, does the bonus attack deal d4 or d8?
Crawford replied:
@ArtificerAlf The bonus attack would use a d4. I like it: druid master of polearms!
The fact that Crawford did not say anything else of their interaction also supports the use of casting ability instead of STR for the attack and damage rolls.
Based to the description of both the feat and the spell the dice used for the attack is a d8, the language used is extremely specific and after reading it carefully you can notice how the two description match perfecly.
My best bet is that Crawford misread Polearm Master (the d4 is called specifically "weapon damage die" as in the description of shillelagh, he believed it was written just as "damage die"; In that case the dice used for the attack would be a d4 and no wander, it's really easy to miss, I had to check it more than once to be sure myself).
All the more the interaction becomes really confusing in practice: how exacly should shillelagh work, just some attack do damage from the magical aura of the spell? The enchantment is weaker based on the way you attack? Or some part of the staff are just not affected?
If they think it's right to use a d4 it's really fine, it barely makes a change (a sad ~1.5 dpr increase in a combat, in the rare occasion you are using both the action and bonus action to attack), and if so they should really make an errata and change the text of the feat and especially explain how the interaction work in practice in game.
For now the manual is clear on using the d8 and for me it makes way more sense practically but if you want to use this interaction, talk to your DM before the session and in the end it's not a big deal, flavour the spell as you like and have fun with it regardless of the dice used.
p.s.: nobody in the right mind would use this on a druid in a challenging campaign, their actions are way too valuable already and going melee is too risky for concentration, the only way this can be an interesting option seem to be a 'SAD' Ranger meme build but it's way far from the optimized standard of damage
