Mods

Client Mods are modifications to your Minecraft program (Alpha or Beta). They are not custom clients, and require modification of minecraft.jar. Like high resolution textures, they only work with the launchers, and won't work if you play  Alpha through a browser.

The most common form of client mods are aesthetic and/or functional. However, some client mods add content to the game, and others fix issues with the game. While some do work on SMP, mods that add content, and thus  could be used for griefing, do not work, as the server will ignore  custom content.

This list only contains entries that cannot contribute to griefing or cheating in multiplayer.

Mod Safety
You must never try to install a mod into the game unless it has been updated to exactly match the version of the client you are running (see the Latest Compatible Release column below). Even a slight variation could be enough to cause problems, including crashing or corrupting your saves. Mods are typically updated on an irregular schedule shortly after the game itself is updated (though this depends on how quickly the mod author or maintainer gets to it), so there will always be a period following a new release when no mods are safe to use yet.

Mods that only alter the behavior of existing items, blocks, or mobs are fairly safe; they should typically be able to be installed or uninstalled at will, without corrupting your save (although if you have structures that rely on behavior introduced by a mod, then they won't work without the mod installed). Note that this includes mods that add crafting recipes without adding new items/blocks.

Mods that introduce new items, blocks, or mobs are not safe to uninstall. Once you have saved a new item/block/mob into your world, you must never play that world without having the mod installed, or you risk crashes or corruption. (Should you wish to uninstall the mod, you must either abandon your previous saves that used it, or ensure that you destroy all of the new items/blocks/mobs in your world before removing the mod.) Also remember that updating the client counts as uninstalling the mod (unless you immediately reinstall the updated version of the mod).

One final note is that multiple mods may conflict with each other; certainly if two mods need to alter the same files, then they are unlikely to be compatible. (The ModLoader provides a means for certain types of mods to co-exist peacefully.)

Other Information
This lists may not always be completely up-to-date. Always check the forum posts of each individual mod. If you don't think this page is up-to-date or correct, please do update it here. Alternatively, there are other worthwhile links that provide collected information.
 * The One Stop Mod Shop forum thread is often updated with each mod that is compatible with the newest version, and also includes any new mods that may have been created.