Community Day Pokémon have, more or less, followed a pattern. Outside of Pikachu and Eevee, every single one has been a 3-stage Pokémon whose Shiny form has yet to be released. Similarly, as every Pokémon gets a special move on their Community Day, having a “Signature Move” helps a Pokémon’s odds.
Barring any Generation 5 Pokémon, Niantic is actually starting to run out of 3-stage Pokémon to use for Community Days, so here are a few possibilities for what's coming up, ranked by how likely each mon might be to get a Community Day!
Other than having a very cool shiny, Rhyperior also has a signature move in Rock Wrecker which has yet to be implemented. While the Rock-type attacking niche is currently ruled by Tyranitar and Rampardos, a Community Day event could absolutely make Rhyperior stand out!
The Porygon line has a number of signature moves: Conversion, Conversion2, and Tri Attack. Additionally, it's one of a relatively limited number of species that gains access to Lock On, which was recently found in the game files. While the Klink line, already confirmed for the first wave of Gen 5, also learns this move, Porygon could certainly get it too. While Niantic has shown a willingness to make trainers use Sinnoh Stones on Community Days, the limited accessibility of Up-Grade (necessary to obtain Porygon2) could make this one iffier.
The elusive Garchomp is all but guaranteed to get a Community Day, being Gen 4's Pseudo-Legendary, similar to Dragonite, Tyranitar, Metagross, and Salamence before it. At this point the question is less "if" and more "when". Garchomp is likely to either receive Earth Power (a surprising exclusion on its release) or possibly even Dragon Rush (a powerful Dragon-type attack that was implemented in Generation 4). Earth Power would leave Garchomp as the most powerful Ground-type attacker in the game, beating out even Groudon!
There are a number of moves Garchomp could receive on its community day, from Dragon Rush (a powerful Dragon-type attack introduced in Generation 4), Earth Power (which would make Garchomp a more powerful Ground-type attacker than even Groudon), or possibly even Dual Chop (which could potentially be implemented as a 3-bar Dragon-type charged move).
All that said, don't expect this event soon. Gible's rarity makes it an incredibly exciting Pokémon, and one that's probably selling Niantic a lot of incubators. The longer it stays rare, the more the hype will build for its eventual Community Day.
Always a fan favorite, Flygon notably did not receive Earth Power when the move first came out. This could make it an easy pick for a Community Day event, but the fact that it shares both the Dragon/Ground typing and two possible moves (given that it also learns Dragon Rush) with Garchomp, the fact that Garchomp is so likely to get a Community Day may lower Flygon’s chances.
With its shiny getting accidentally released for like an hour at Chicago GoFest, there's always a chance it comes back as a Community Day Pokémon. While its signature move, Kinesis, is only a status move in the main series, Niantic’s willingness to set up Charm as a powerful damaging move does bode well for this supergenius telekinetic.
Spheal: a round, wonderful friend, without access to its signature move Ice Ball. Walrein was historically a top Rayquaza counter way back before anything good became available. While Spheal does seem like a bit of an underwhelming choice for a community day, Niantic did give us Slakoth Day, which absolutely nobody was asking for, so you never really know.
Not Very Likely
Victreebel, Jumpluff, Shiftry, Vileplume/Bellossom, Exploud, and Staraptor are all 3-stage evolutions that do not yet have shinies. However, they all lack Signature Moves, and honestly aren’t that exciting. If these Pokémon were to receive shinies, the releases would more likely be attached to events.
Wurmple, while technically a 3-stage evolution, is probably more likely to get its own special event. We miss you, Party Hat Wurmple. Please come back.
Generation 5: Looking to the Future
While there are a lot of unknowns to Generation 5, we can make a few safe guesses.
As per the pattern of the last few years, the three starters (Serperior, Emboar, and Samurott) are absolutely going to get Community Days, learning Frenzy Plant, Blast Burn, and Hydro Cannon respectively. If the pattern of Gen 3 holds, they may also receive a permanent TMable move as well! Hydreigon, being Gen 5's Pseudo-Legendary, is also all but guaranteed to receive a Community Day.
Haxorus, while not a pseudo-legendary, is an iconic Dragon-type Pokémon with an utterly gorgeous shiny which would make for a very exciting event. Krookodile could receive its signature move Power Trip. Vanilluxe, everyone's favorite example of how Nintendo has totally jumped the shark, would be a super cute and thematic Community Day for Summer of 2020. Chandelure has a couple of iconic moves in Hex and Flame Burst, which could function as Community Day options too!
Breaking the Pattern
As stated above, the majority of previous Community Days have been 3-stage Pokémon. But what if Niantic were to break the current paradigm and enable 2-stage Pokémon? This would enable a number of new Pokémon to receive their signature move via Community Day, such as Barrage Exeggutor or Night Daze Zoroark.
Community Day Reruns
Waiting until December to get another shot at Last Resort Umbreon or Meteor Mash Metagross last year sucked. Given that this year we’re not even sure we will be able to get those moves back, it’s been even more stressful. As a way to (at least partially) remedy this, Niantic could decide to just do repeat Community Day events, perhaps in batches a la December 2018’s Community Weekend. For example, one month could get “Meteor Mash Metagross and Smack Down Tyranitar Day.”
Who’s to say that a Pokémon currently having a shiny available means it can never get a Community Day? Weavile can’t learn Beat Up, and Hariyama can’t learn Smelling Salts. Despite already being shiny-eligible, Niantic could absolutely decide to give them Cday events with their signature moves.
Alolan Sandshrew and Alolan Vulpix are currently only accessible via eggs, mostly. At 2019’s Chicago and Yokohama Go Fest events, they showed up in the wild. This sets at least somewhat of a precedent of Niantic putting egg-only species into the wild for an event. Allowing typically Egg-limited Pokémon could enable Lucario getting Aura Sphere, or something like Happiny, Tyrogue, or Munchlax cdays!
In the end, the current Community Day paradigm is completely arbitrary. Not every special move has to be a "Signature Move", not every Pokémon has to have 3 stages. Niantic has complete freedom to do whatever they want with these events, and maybe, in the future, they will!