I seriously made an account just to post on this thread. I'm not sure how that makes me feel...
Anyways the worst MU's for Doc that I've found at least to be on FG are
Rosalina & Luma,
Shulk,
Marth/Lucina,
Sonic,
Sheik.
Rosalina:
Is at the front mainly because of Luma, every single one of Doc's hits on Luma first creating zero stun on Rosalina which means she's still able to act even as you throw out a f-smash or Bair, even then if your hit connects with Rosalina first Luma is still able to Up-smash your approach. All of Rosalina's aerials outrange Doc or trade with him almost nullifying his Bair game. Doc has a fairly reasonable MU potential against everyone but a good Rosalina main will always beat a Doc.
Don'ts: Don't ever underestimate the range of Rosalina's aerials from her Nair to Dair if it doesn't look like its going to hit you, it probably will along with her charge shot in which she launches out Luma, this move does kill.
Shulk:
If you're playing against a spacing Shulk and you're not using fox trot shielding or some other method of approach you're going to get Fair'd, Nair'd to death, although there are several openings to punish with your Bair, a surprise Fair or even Nair.
For instance if Shulk is prepping a F-smash best punishes due to ending lag on all of Shulk's smashes in my cases are: Dash, Pivot, Short hop, Bair. This usually can be done before the move goes off and if successful shorthopping over any or all of his smashes.
Don'ts: Don't dash attack, ever. Nair and Fair will punish it every time or even a possible Fsmash. Approach using pills and Nairs.
Marth/Lucina:
Not going to go into this one because the difficulties have already been listed above with this MU.
Don'ts: Don't use your F-smash on a counter-heavy Marth/Lucina, it will kill. D-smash and foxtrotting shielding will make it a little bit more bearable along with your pills, if s/he is spacing you out space them out with retreating pills until they make a mistake.
Sonic:
This is a rough MU especially on FG where this character shines due to lag where timing Doc's cape becomes a chore rather than a tool. A well placed Cape can ruin any Sonic but the timing is very short, not only that but Sonic's side-b rush into U-air is almost guaranteed without some kind of DI. Sonic is able to edge-guard Doc like nothing as well using his spring offstage to knock Doc out of his jump/recovery. Thankfully the Tornado is able to plow through the spring, but the timing [with FG] is horrifying. If you're playing against a Sonic with a really bad internet connection there's usually a reason for it.
Otherwise this is not the worst MU for Doc.
Don'ts: Don't let go of your shield too early, learn the timing along with how many hits each of Sonic's attacks dish out. Don't use just your Up-B to recover, the Tornado is tricky to use at times but it catches A LOT of people off guard when you recover from a place that you normally wouldn't with Up-B.
I'm not a master by any means, but these are just some of the MU's that I've found and how to do deal with some of
them. Surprisingly Diddy Kong isn't Doc's worst MU, its one of his better ones.
Sorry for spilling out so much stuff, just thought it was an interesting topic as a PhD main wanted to put in.
Edit: I forgot OoSO such as, Shield, short hop, up-b. This works in a lot of cases especially on Sonic and Sheik.