Screw-gates rule. Carabiners that "auto-lock" are great in the store, but hideous in real use most of the time. Do NOT use any neato self-locking carabiner in winter, doing so will only annoy the hell out of you and could be dangerous if it permanently freezes shut. It's easy to operate a screw-gate even with reasonably thick gloves on, but try to set up your belay device with an "auto-cluster" 'biner and you'll likely have to remove your gloves. You'll definitely have to remove your gloves if there is even a smidgen of ice or hard snow stuck into the mechanism, which there often is with auto-cluster biners.
If you're doing a multi-pitch route and have to deal with a minimum of three locking carabiners (two on your ATC to belay the second, one on it to belay a leader, plus all the other uses for locking carabiners) per pitch you'll likely stop using auto-lockers after a few pitches due to the annoyance level. They are also a huge waste of time, even if they don't freeze shut. Figure an extra ten seconds per biner to sort it out with gloves on and that's an extra minute per pitch (six locking/unlocking cycles per pitch minimum). On a ten-pitch route that's ten minutes, or more if even one of the damned things freezes shut.
You're also more likely to drop your belay device or yourself--trying to open the auto-lock with gloves on while holding the rope and your device is an epic. Many "auto-lockers" totally cease functioning in wet and cold conditions, or at least stop locking properly. With a basic screw gate these problems are almost always pretty quick to solve (whack the sleeve gently with a pin or ice tool, worst-case scenario hold the carabiner in your hand for a few seconds until it warms up), but with an auto-lock I've had to untie or cut cord to get the problem sorted before I froze to death or became so angry I hurled the whole mess off the cliff.
In summer I still like basic screw gates over auto-lockers, for many of the same reasons. It's common when doing multi-pitch routes to have to sort out various pieces of rope, cord and the anchors; this is near-impossible to do while holding the auto-locker open with one hand, rotating it, etc. I know many schools like to teach or guide with auto-lock biners, but if you're not a student then auto-"irritaters" are just that. Hint--if you're hanging at a belay and tighten your screw gates while hanging on them they will be hard to open. Loosen the screw gate before unloading the biner.
I do have a couple of smallish "pear" biners for belaying with, the rope runs smoother and belay devices tend not to get "stuck" in the angle. I keep these biners with my belay devices. I also have one weird old carabiner with a wide "spine," that one works well when using it with an ATC or other auto-block, if the ropes not totally loaded it's easier to feed slack by turning the 'biner.
Some people use the big auto-lockers for fixing ropes, as a screw-gate can be opened by contact with the rock. When I'm climbing for myself with friends I generally use two non-locking carabiners with the gates reversed--nothing beats that, and it's still manageable if I have to move it around. For film or industrial rigging I use super-heavy steel biners, often auto-locks. In this situation I'm not pressed for time (even if I am), and want something ridiculously bomber in case of a very weird of very high-load situation, something that happens occasionally when working with power winches or hauling systems. I also won't be watching the rigging sometimes, so overkill is good. The auto-lock carabiners also deter people from messing with them...
General use carabiners:
Wire gates rule for most types of climbing. Anything else is too prone to blockage with ice and snow, and also reduces the span between the gate and the "spine" of the biner more than necessary. Most of the time I use the smallest, lightest biners I can get my hands on (the BD Oz or Neutrino). I carry a few huge wire gate biners to rack pins on (screws are racked on ice clippers). Sometimes old-school ovals are nice for racking on or clipping pins in confined locations, but an Oz will fit just about anywhere an oval will and some places an old-school oval won't. Aid climbers and other bondage fans also like ovals; enough said then.
The difference a rack of Oz biners (28g/1oz) makes to the total weight of a big rock rack can add up to several pounds. For example, if you're doing a big rock route you might have 48 or more carabiners between you and your partner (biners on two sets of cams, quickdraws, biners on slings, etc.). If you use an oval or other heavy (2 oz or more) biner your rack will weigh at least three pounds more. That's 1.5 liters of water, a six-pack of beer, six hamburgers, a huge belay jacket, you get the idea...
For sport climbing I like draws with no "gate catches" on the bolt end, and a big-mouthed wire biner on the rope end. Cleaning draws off overhaning routes with gate catches sucks, they catch on the bolt. Big-mouthed wire-gate biners are really easy to clip the rope into, which is what you want. As an aside, any route where your belayer can still accurately toss you a beer after you've lowered off is a slab. Just wanted to clear that up.
I have a silly quantity of carabiners that I've gathered over the years. I just really like having the right tool for the job. Not to brag, but I probably have 200 biners and use 'em all. Even with all that I don't have enough to have the exact perfect setup for everything. I find myself building sets of quickdraws for alpine climbing (Oz biners on both ends, skinny long slings for the webbing), then rebuilding a set for sport cragging (Positrons and Live-Wires), and then yet another for mixed climbing (fat webbing that will take the abuse of tools stuck through the biner) with positrons and Live Wires, and then yet another for long free rock climbs... If I could only have one carabiner for everything I'd go with all Neutrinos, they are big enough to use for most things and light enough not to be a waste of metal. Hey, that's it, using fat carabiners isn't just heavy, it's an environmental waste too.
If I'm going new-routing I try to always bring a few "POS" biners to leave for raps. These 'biners will work for other stuff, unlike rap rings or Maillon Rapides (quicklinks), and make pulling the rope a lot easier.
That's how I see it. There are a thousand safety discussions to be had about carabiner choice, but this is what I've found works over the years. I've broken three carabiners over the years, and seen another half-dozen fail for one reason or another. Good carabiners are important, and remember that two non-lockers is better than about anything else for high-stress or "weird" situations. I've never seen two non-locking biners with the gates reversed fail.