There are 29 rings in NetHack, and only 2 ring fingers on each player. The rings have varying utility, hence a strategy of which ring to wear must be devised. Rings do cause extra hunger (some excessively) but after the mid game food should not be an issue, meaning there is no reason not to make use of the many ring extrinsics. The rings can be divided by how useful they are. Note, however, that even "useless" rings can have some use (e.g., taming a pet with a meat ring, or wearing a ring of hunger to make it easier to eat multiple giants on a level).
Normally useless rings:
Mostly useless rings:
Useful but supersedable rings:
- protection from shape changers
- cold resistance
- gain constitution
- gain strength
- increase accuracy
- increase damage
- poison resistance
- see invisible
- shock resistance
- fire resistance
- slow digestion
- teleportation and teleport control
- polymorph and polymorph control
This division is somewhat arbitrary, and many might disagree with the particulars; some prefer to wear slow digestion at all times, while others consider free action a nicety rather than a necessity.
The useless and mostly useless rings are usually discarded on sight, or stored for polyfodder. The third category of rings are very useful in the early game (a ring of poison resistance can be a life saver if you lack the intrinsic), but are superseded either by gaining the intrinsic or becoming as strong/tough/accurate as it is possible to be. Protection from shape changers is partially supersedable (for werecreatures, not protecting from chameleons etc), and only by wielding Werebane, but a cloak with level 3 magic cancellation all but nullifies the lycanthropy attacks of were-creatures.
Remember that you can polymorph into a metallivore and attempt to gain intrinsics by eating rings (a decent 1/3 chance per ring). You can accumulate many minor but useful intrinsics using spare rings this way (which can be in turn obtained via, for example, a bout of pudding farming), thus leaving your hands free for rarer or inedible rings.
More importantly, in the early to mid game, your choice of ring is limited to what you can find, so this page will discuss what to do when you actually have a choice.
The useful rings are good candidates to keep in your inventory at all times. A ring of protection looks less attractive later on when your AC is already below -30, as does a ring of warning, since several classes gain warning either as an intrinsic or as a property of their quest artifact, or failing that an equally useful method of sensing monsters at a distance.
Regeneration is very handy in a prolonged fight, or when encumbered. Slow digestion is potentially a life saver for early spellcasters or those who are conduct-restricted in what they can eat (which mostly means vegan or vegetarian monks), but it is by no means compulsory later on (and indeed can be a hindrance if you come across a corpse that you really want to eat but are satiated; this is also the reason why rings of hunger are not in the completely useless category).
Teleport and teleport control are technically supersedable, but tengu are rare, so it is not always possible. A common strategy is to carry only the ring of teleport control, having purposely gained teleportitis. This is not a waste of a ring slot, since the extra effects of rings are most useful in large or difficult fights, and these occur disproportionately on non-teleport levels where you don't have to worry about teleportitis.
Polymorph and polymorph control are very advanced rings which, although useful, are best used in controlled circumstances such as a locked co-aligned temple with nearby stash. Hence they are not particularly useful to carry around all the time, and certainly not to wear all the time. Polymorph control, worn by itself, can spare you the headaches associated with polymorph traps (such as armor destruction) if you do not already have magic resistance.
The essential rings are not absolutely necessary to ascend, but they make it a lot easier. Free action protects you from paralysis when fighting high level spellcasters (and so should be worn throughout the end game). Levitation allows you to easily cross several water levels, as well as the planes of Air, Fire and Water. Finally, conflict causes you 50% extra hunger, but will save your life many times over in a heavy fight, as well as give you an easier, faster path through crowds of monsters.