The existing answers to this question are focused on very specific knives that are favored. I will try to give more of a general approach and philosophy.
As with all outdoor equipment and techniques, we first need to identify what problems we are trying to solve and what some of the design tradeoffs & constraints might be.
- As identified in the question, the main uses for a knife when climbing are cleaning/building rappel anchors and cutting stuck ropes.
- Contrary to what Hollywood portrays, needing quick access to cut your partners rope is basically a non-existent worry. There are only a handful of instances and freak accidents in mountaineering history where cutting someone free was necessary. Compare this to a sport like whitewater kayaking or sailing wherein becoming entangled in a rope is a life-threatening emergency that can only be remedied with rapid access to a knife.
- Other possible problems & use cases for a knife when climbing:
- General outdoors uses (10 essentials), cutting salami/cheese at lunch, ...
- Getting a long beard / long hair / loose clothing / loose climbing equipment caught in a belay device while on rappel. An ounce of prevention is better than a pound of cure and there are other safer self-rescue options. If you employ a knife in such a situation, be very cognizant of the fact that you're using a knife next to a life-supporting rope.
- Climbing ropes and other textiles are generally relatively simple to cut, especially if you can (safely) put some tension on them. Even just rubbing a tensioned rope back and forth over a rock edge will get you through a rope in a few minutes of effort, as will sawing a tensioned line with a sacrificial cord. Knowledge, improvisation, and practice are cheap, lightweight, and easy to carry.
- Once again, climbing equipment is sensitive to being cut (along with you, a spongy meatbag). Having an errant knife cut you or your equipment is likely a larger hazard than not having a knife.
All of the above lead me to the following preferences:
- Light is right! Especially when a knife will have, at best, sporadic and non-emergency uses. And the largest thing I might need to cut is a 10mm (3/8") rope. Or my lunchtime salami.
- Any knife should be secured safe: a folding knife that locks closed, a folding knife taped shut, a very secure sheath, etc. Bonus considerations for not being in a pants pocket or dangling on a harness, wherein it would create an emergency were it to come unsecured.
- If I have a knife, I want a functional, reliable knife: quality steel, at least partially serrated, sturdy, good ergonomics.
Given these parameters, here are my preferred options. You might weigh the above criteria differently and come to a different conclusion that is better for you---good!
- In well-established areas or when single-pitch climbing, no knife at all. Worst case, I can deal with cutting a rope or anchor material with a rock. Alternatively, I have a knife in the top lid of my pack at the base of the climb.
- On adventurous multi-pitch climbs or mountaineering, I keep a small folding knife, securely held closed with multiple wraps of climbing tape, inside a zippered pocket of my chalk bag or in the top lid of the pack. In my case, it usually happens to be a Spyderco Ladybug---the smallest, lightest knife with very good steel and surprisingly good ergonomics that I am aware of for a non-exorbitant price.
- I greatly dislike the climbing specific "knife + thing" multitools, e.g., Trango's knife + nut tool or Grivel's ice v-thread tool + knife (more like loosely secured razor blade). I think they create a new hazard, are horrid to use as a knife, and are overpriced. Similarly, I am wary of any knife sold by a climbing company---their expertise is in designing and manufacturing climbing equipment, not knives. Especially when their "climbing specific knife" is a non-locking (open or closed) folding knife with a carabiner hole, that can come open and stab you while it's clipped to your harness.
- If you insist on having the knife clipped to your harness, many knives have a lanyard hole---a small loop of cord gives a tidy clip in spot. Just make sure to tape it shut!
As you can see, my personal calculus is that knives aren't all that mandatory in a climbing context (and I say that as someone that almost always has a knife in my pocket in my daily life). The problems that they can solve generally aren't emergencies requiring split-second access and an unsecured knife can introduce a whole host of worse problems. Perhaps your personal calculus is different---perhaps you are worried about the handful of strangulations caused by loose slings / shoulder racks getting snagged during a fall; perhaps you enjoy letting your long hair billow in the wind while on rappel. If so, you might put a high-emphasis on quick, one-handed access. Just be sure to account for what the other dangers are lurking and make examined decisions.
Edit: More options
- A harness-clipped knife could also be held shut with a heavy-duty rubber band (periodically inspect & replace). A section of a bicycle inner tube also works great.
- One ultralight option: remove a single-edge razor blade from its packaging, fold a piece of climbing tape over the blade, and then tape it securely to the inside of your helmet (give some thought to the positioning). Weighs virtually nothing, stays out of the way, impossible to forget, but is a bit annoying to use as a knife, you have to take your helmet off to get to it, arts & crafts project to stow after using, marginal risk of cutting your scalp if your helmet takes a hard impact (works better with hardshell "brain bucket" helmets).
- Bring along a dedicated short (~18") piece of skinny cord (~2mm) with a simple loop knot in each end. When needed, clip a biner to each loop and saw back and forth over a tensioned rope to cut it. Works great for normal climbing textiles, weighs virtually nothing, easily stored, zero chance of accidentally cutting yourself or your gear. Can falter against some textiles (kevlar/aramid), less precise and slightly slower than a proper knife, almost useless for other knife-type tasks. Can be improvised with, e.g., a shoelace or backpack drawstring.
