Can other mobs spawn in the vicinity of a mob spawner?

I have a simple experience farm with a zombie spawner (water pushes the zombies into a trench where I can beat them to death). I'm wondering if any other mobs can spawn in the dark room that contains the zombie spawner, or is it some sort of exclusion area for other mobs?

I'm mostly worried about going afk near the beating hole and having a creeper spawn and blow up next to me.

There is a safe radius around the player where creatures can not spawn. Normally (depending on how big of a 'trap' you have built) the player's radius includes a dungeon spawning trap. The Dungeon Spawner is unique in that it ignores this normal requirement for spawning, which is why they keep appearing in front of you.

So, if your no-spawn radius includes the dungeon then you are ok. Also of note is that dungeon spawners will spawn creatures in mid-air and above water, normal creatures can not do this because of the spawning rules. Again, depending on the design of your trap it should be inherently safe.

However if wherever you are standing can be reached by just walking or 1 block jumps or multi-block drops, be aware that something can spawn outside of your safety radius and eventually find its way to you. I suggest only going AFK if you are completely sealed off :)

I agree with James's answer, but I wanted to work out the specific figures.

• Spawners are active within 16 blocks (Euclidean) of the player, and may spawn monsters up to 5 blocks (Chebyshev) of themselves [ref], for an absolute maximum player-monster distance of ∼23.07 blocks.
• The exclusion radius around the player is 24 [ref].

So, from these figures, you cannot have other mobs spawning as long as you are actually at the spawner.

Caveats:

• If the chamber around the spawner has dry ground in a larger area (by two blocks) than the spawner's area, then it could spawn other mobs. Therefore, don't exceed the 8×8 (or 9×9 if you don't like pondering coordinate axes) optimal size.
• As James pointed out, if the spaces connected to your idling location are not perfectly lit, something could wander in from there; therefore if you don't trust your lighting skills you may wish to put doors around your idling location.

These conclusions agree with my own experience: I have had zombies and creepers in my skeleton farm (my design has some dry ground in the spawner chamber), but only ones which spawned before I arrived at it — never any more once I start idling at the output.

You specifically mention being worried about creepers. One way to fully protect yourself is to make sure that the mobs do not have a line of sight to you — creepers will not explode and skeletons will not shoot unless they can see your face (and not even through glass). Therefore, a simple safety measure is to confine the mobs such that blocks are in front of their faces and you hit them in the legs.

Also, you can protect against spiders in your zombie/skeleton trap, or creepers in your spider trap, by making sure there is a passage only large enough for that type of mob. (This doesn't help with creepers in zombie/skeleton traps since they are the same size.)