I really hope contests appear again in Omega and Alpha! I loved earning ribbons, and making Poffins! Without contests, berries aren't nearly as useful as they used to be...
They were great in Ruby and Sapphire. It offerered new strategies and training. It was a fun mini game.
After R/S imo they went downhill. Rules were changed, and pokemon musicals were boring and pointless
I'm going to have to agree with Din. R/S/E did a great job with the contests…they were different enough from the normal game that it felt fresh, but similar enough that it was comfortable doing them (I do wish they'd've put in the anime "battle your opponent in a stylish manner" bit, but with the technology even today I don't think that'd be possible to code). D/P/P added a couple of things I really didn't like to contests, dress up and dancing. The dress up bit felt forced and it was really annoying. Unlike before it wasn't clear which items would be seen as "sharp" or "cool"…you could guess reasonably well but it wasn't easy…and I simply didn't like how the items didn't look good on the Pokemon they were attached to…in addition, it was a nod to people who dress real life pets up like that, something I without exception find ludicrous. The dance competition was another gimmick, but a somewhat more successful one. I did enjoy it enough to say it was a good idea, but it should have been its own minigame or sidegame, not part of a Pokemon Contest, which had been established as "what Pokemon has been groomed for this situation and has moves to support it?" as dancing was purely up to the player's reactions and short term memory. The Pokemon had no reason to even be in the dance bit…if you had the trainers doing it it'd've been exactly the same game. So, while it wasn't as bad of an idea as the dress up part, it still didn't feel like part of the contest experience. After that they tried to transition to musicals, which were a huge flop.
So, what do I think ended contests? They tried to change the formula fans enjoyed too much. They went from "did you put time and effort into making this Pokemon the coolest/smartest/cutest/most beautiful/toughest looking Pokemon compared to the others, and did you give it a moveset that further exemplifies this attribute?" to "Did you find enough items that fit the theme to dress up your Pokemon, do you as a human being have good reactions and short term memory? Oh, and do you have a good moveset on the Pokemon you entered with?" It just got too clunky to continue. If they show back up in their original format in OR/AS I'd be thrilled. Just don't make it a D/P/P format or anything like it, or else I'll be rather disappointed.