Does the magic missile simply bypass Mirror Image, since it has no attack roll, and thus it's not an "attack"?
In 4e it would have been as simple as it not 'hitting' ("simple" is not exactly the word, MM was updated to an effect instead of hit line, and a cascade of changes followed to make it work). 5e's natural language means that the idea of being hit when attacked or hit by a missile might well be separable from the mechanics of an attack roll and hit (or not), even though both spells use the word 'hit' they might be seen as using it differently (Mirror Image in a jargon sense of a successful attack roll, Magic Missile in a fluff sense of inflicting damage automatically). So, you've plenty of room to rule either way, even if you're trying to stick close to a literal interpretation.
Would you allow them to target 1 missile at each duplicate in an effort to destroy them? (as a 2nd level spell, you could target 1 missile at each image you see, only 1 of which would be the real).
I'd be inclined to, as they are simultaneous so the difficulty in 'tracking' illusory duplicates is not an issue....
I'm pretty sure this will come up. How would you handle it?
However, I also tend to like to just rule how I've handled things in the classic game. The classic Mirror Image had language about the images showing the same injuries and reactions as yourself, so if you were caught in a fireball, they'd all look a little toasty. And the 5e magic missile makes it flatly impossible to tell an image, so it presumably acts similarly. By that logic, an attack like magic missile could have targeted you (since the caster can see you, even if he's not sure which location you're in) and it would just look like all the images have been hit and damaged the same as you.
Part of the classic feel of magic missile is that it's auto-damage, letting it unerringly strike the target in spite of mirror images, but not affect the images, either, sounds fine, and it lets both spells be cool.
The more obvious mechanical interpretation would be to assume you can't target images intentionally, that you have to target the caster, then roll that d20 for each missile... a problem with this is that, since the missiles are simultaneous, it suddenly really matters which other image they hit, and the spell doesn't provide for that, so you're back to making a ruling, another is that it may seem strange to say that the caster of the MM 'can't' target each image he can see, but must target the caster of the images, then determine randomly whether he targeted an image,
which he wasn't allowed to do in the first place. ::shrug::