Is it possible for a Mob to spawn *right next to* the player in Minecraft?

Unless other changes have been made to the spawning code, enemy mobs can't spawn within 24 blocks of your current location. The exception to this is of course mob spawners, but that's a different story.

They can't spawn right next to you; you have a 24-block radius of safety. They can approach you afterwards, but that's different.

In addition to the other answers, a recent change (pre-1.6.1 13w17a) has introduced a feature where a zombie can summon more zombies when they are damaged regardless of damage source, which ignores light level or player range.

Therefore, specifically for zombies, it is possible for a new zombie to randomly spawn near players within the player immobility range if a zombie was damaged and a new one randomly spawned.