Are there any handicapped ponies (outside of complications due to age) shown in the show? As far as I know, there aren't, and the only ponies/creatures with amputated limbs/prosthesis/broken horns are in the movie, and there is no mention of them in the show afterwards. So can people reply with examples of canon characters/background characters with one of these traits? I do know that Crackle may be mentally handicapped, but given a lack of speaking lines, I'll put her on the shelf for this.

Derpy is one with the eye problem but I don't know any others.

Scootaloo's under-developed wings?

Rainbow Falls had a handicapped pony I believe who had some sort of gadget attached to his back legs for movement.

Correction Steller Eclipse from "Trade Ya" walks with a wheelchair cart for his hind legs.

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There's Stellar Eclipse from the episode Trade Ya':

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Derpy's eye problem is not what I had in mind. I meant stuff that would be glaringly obvious, like an amputation, Down Syndrome, and the like.

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A few others come to mind:
Diabetty's morbid obesity.
The wheelchair pony from Trade Ya
Grandpa Gruff who is blind in one eye.
Mentally ill Screw Loose
Tempest Shadow's broken horn.
AU Rainbow Dash's mecha-wing.
Spike. In general. lol jk

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Ah, but he doesn't have something obviously wrong with him. Out of context, you would assume that he had on the wheelchair temporarily, given how his back legs are the same developmentally as the front, when the front should be disproportionately larger.

In the episode “where the apple lies” you see a male Unicorn with a bandaged horn. The top of the horn is flat so it safe to say it’s broken.

There were a lot in Leap of faith

Theres also Screwloose she is the barking pony

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Well, his legs aren't in casts which indicates they aren't healing from anything. I would assume they just don't work properly. It could have been due to an injury as an adult.

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Unlike Stellar Eclipse, this one couldn't afford health insurance. Apparently her name is Jinx.

I guess Mr. Mouse started asked his wife why the kid looks nothing like any of them and that's why he ended up in a wheelchair.

Also, the Sombra timeline had Rainbow Dash with a prosthetics wing:

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Aren't they the same size as Rumble's, though? And Bulk Biceps', who is an adult? And filly Rainbow Dash's? And pretty much every single standard-issue pegasus foal we've seen? It's kind of weird, too, how Rumble is the only foal we've seen get up to tree height flying, and throughout Sonic Rainboom there didn't appear to be a single foal up in Cloudsdale...

*ahem* Sorry, trying not to ramble too much. Back to the main point.

You could argue Bulk Biceps has underdeveloped wings. I don't think anyone has the heart or stones to call him handicapped to his face, though.

The character from Trade Ya was an OC, added as something or other with regards to the Make A Wish Foundation. Not sure how official you'll want to consider that, then, but it's on-screen, so... make of that what you will.

Flurry Heart could be handicapped, or at least require medical attention. Her wings are so large, she might develop back problems during a growth spurt.

On the top of my head, there were a lot of injured ponies in "Leap of Faith" Silver Shill pretended to be handicapped and then healed.
Then there's Stellar Eclipse from "Trade Ya" a fan OC in what I for lack of words will call a quadraped wheelchair.
In one of the alternative histories caused by Starlight Glimmer we see Rainbow with a metal wing. Could be armour of course but since her other wing is normal I think it is a prostethic.
And that one hospital patient in "Read it and Weep" who barks like a dog and is handled by the hospital staff.

There are plenty of cases where ponies get temporarily handicapped due to injuries, but they tend to heal after a while. In "Read it and Weep" both Daring and Dash injured their wings and couldn't fly. Then Twilight was put in a wheelchair in "Feeling Pinkie Keen" due to her injuries.

Apart from that one barking dog there doesn't seem to be any mentally handicapped ponies, at least not bad enough not to function. Then again they have never visited a mental asylum either. The show flirts a lot with mental problems but all the ponies seems to be dealing with them well enough to function.