I've taken half an hour just looking through the auction house, and I have taken my own experience and various forum posts into account, and these are my findings:
The maximum number of sockets per equipment type varies. It seems to be independent of item rarity (magic and rare items, legendaries are something special). Instead, it seems to depend entirely on the body type of the item.
These are the maximums I was able to identify:
First of all: Shoulders, belts, gloves, and boots can't have
sockets at all (more on that later).
Helmets, rings, amulets, weapons, and offhands:
Legendary items are a special case: They always have a certain fixed amount of sockets per named item, but just for completeness: The maximum number of sockets on legendaries I could find was two.
There's something I have to add about Shoulders, belts, gloves, and boots:
Even though there seem to be no belts with sockets, it is theoretically still possible that belts can have sockets, and simply nobody has found one yet. I come to this conclusion from a simple fact: Shoulders, gloves, and boots have no search option for "has sockets" in the auction house, indicating that it is indeed not possible for them to have one. Belts, however, do have this search option. It is entirely possible that Blizzard just left it in by accident, though.
Another interesting fact about items and sockets:
Items have a fixed amount of random properties (this can differ, even for same rarity and body placement). When an item is dropped, or identified, the contents of these properties are randomly rolled out. Sockets, no matter how many there are in the item slot, take up exactly one of these random properties.
This make for an interesting conclusion: If you're unlucky, sockets may take away important stats from your item. In general, pure item stats will always be higher than you could at to it with gems. For example, perfect strength gems provide +22 strength. In comparison, items of roughly the same level seem to hit an upper cap of around +60 strength. This means that if you, for example, identify a rare chest armor piece and it finds +50 vitality, +20 experience per kill and +1.2% chance to stun on hit, plus one socket, then that last socket could have been a slot for +50 strength, whereas now you can only try to compensate for it with gems.
On the contrary, however, this becomes a nice bonus when you actually want to stack a certain stats. Items can't roll the same bonus stat twice. If we go with the example from above, you now can circumvent this fact by adding another +22 vitality gem into the socket, thus giving you even more life on the item (good for tanks, for example).
This thread discusses all of this (notice Bashiok's post later in the thread, acknowledging these facts).