Simply put, a Stable is the set of champions someone aspiring to play a certain role would want to have. It's the set of staple champions you'll want if you're going to play role X.
THIS IS A META LIST. All the champions and roles on this thread will be META champions and META roles. I encourage deviating from the meta, as it results in a fun and interesting experience, but simply put, doesn't see play. The meta is what 99% of players will play under and what works. I'd be doing everyone a disservice by suggesting someone play a sub-par champion or a role that isn't commonly seen.
PLEASE know that these are only MY opinion, and I'm more than open to changes if you feel any of these are WRONG! PLEASE feel free to (politely) point out any flaws in my logic or suggest other champions for any role you desire. I'm not a high ELO player, but this is what I've found to be a decent pool of champions for a given role.
Ranked in no particular order
The Solo Top Stable:
1. Warwick - Infinite sustain, can be switch to jungle if it all goes wrong, powerful ult.
2. Cho'Gath - Very safe pick, near infinite sustain, incredibly tanky, and high burst damage along with his kit of CC.
3. Yorick - Another safe pick, great sustain and harass power, hard to counter, and scales well.
4. Vladimir - A good counter to certain champions, again great sustain, great escapibility to prevent tower dives and ganks.
5. Shen - Given his recent buffs, Shen is incredibly powerful. Great CC, amazing tankablility, and a global ult that puts Ezreal to shame.
The Jungle Stable:
1. Shyvana - Amazing clear time, fairly strong ganks (lack of cc hurts), high damage, very tanky, scales incredibly
2. Maokai - Provides some AP and very strong pure-tankability, good to have in your kit
3. Udyr - Amazing sustain, hard CC for ganks and teamfights, great damage
4. Lee Sin - Good clear time, versatile, good counter jungler, strong ganks
5. Skarner - Great ult, good damage, incredibly tanky, good 1v1 and strong late game
The Solo Mid Stable:
1. Ryze - Great damage, low cooldowns, tanky later, scales incredibly, good harass
2. Cassiopeia - Great harass, decent cooldowns, hard CC with ult, DoTs, powerful burst
3. Kennen - Teamfight winning ult, doesn't need blue, great CC
4. Ahri - Great lane phase, good sustain, great escapibility and teamfight presence due to her maneuverability
5. Morgana - Powerful pusher, punishes enemy ganking by taking turret early, great ult and hard cc, strong damage
The Support Stable:
1. Soraka - Global heal, helps sustain carry well, Starcall helps teamfights a great deal
2. Alistar - Great CC, heal to sustain carry, zones particurally well
3. Janna - Shield/Steroid is great for aggressive lanes, ult helps break up unwanted fights
4. Lulu - Great early game, removes enemy carry from teamfights for a few precious moments
5. Nunu - Great attack speed steroid, good harass, sets up ganks well, great ult and strong baiter
The Ranged AD Stable:
1. Vayne - Hardest carry, great late game damage, hard to kill as teams have trouble finding her during her ult
2. Kog Maw - Great against tanky teams, does tons of damage and can be deadly if protected
3. Graves - Becomes decently tanky, great attack speed steroid, does tons of damage, good harass
4. Tristana - Amazing attack speed steroid, initiation/disengagement tool, good early game harass, and amazing late game damage
5. Corki - Great escapibility, very strong early game, abusable ult, great tendancy to shut down opposing farm and get fed with appropriate support
4-10-12: Originally posted
4-11-12: Shyvana section reworded, removed Ashe and Urgot from Ranged AD, added Corki and Graves
4-12-12: Removed Warwick from Jungle, replaced with Skarner, removed Sona (given her nerfs and lack of competitive play) from Supports, replaced with Nunu