things that break that can spoil yer day
Freehub pawls: Dunno about downhilling, but if you're strong and pedal uphill, pawls can fail, leaving you in freewheel mode in both directions. I've killed probably 6 or 8 hubs and freewheels on tandems, and 3 or 4 on singles. To get home, use something handy -- fence wire, toe straps, big zip ties, etc -- to tie the largest rear cog to as many spokes as you can on the rear wheel, turning it into a multi-speed fixed gear. It may torque the wheel a little, and you probably can't climb steeps, but you can definitely pedal most terrain and get home. You can even shift gears, just don't use the largest 2 or 3 cogs. And don't try to coast. Once cleaned Pritchett Canyon in Moab fixed gearing a tandem this way, pedaled right back into town. I carry a couple nylon toe straps on backcountry rides for this purpose (and for making splints).
Steerer tube: Lots of aluminum steerers out there, their owners probably don't pull their forks to peer at them closely at least annually, and they should. Steel steerers also can break. Both usually fail right around the crown race. Worth a look, a busted steerer can be real nasty.
Quick release skewers: They usually break right at the threads where the R side end piece screws on. Sometimes the levers fail. Steel and ti, they both break. I carry a spare rear on backcountry rides, with a 35mm spacer to adapt it to the front. Saved a buddy probably a good 6 or 8 hours of tough hiking last summer by having one along.
The wimpy little allen bolts in many 2-bolt seatpost clamps. I've broken several, and now carry a spare.
Seatpost bolt: It'll break when you tighten it after adjusting your saddle height far from home. I carry a spare. The tensioning allen screw in aheadset systems can be used as a seat post clamp, as it's doing absolutely nothing once the headset adjustment is locked in by tightening the stem.
Rear shifter cable: It'll break right at the barrel end, not a crisis, you can always one-speed it out, but inconvenient. For some reason I have never broken a front shifter cable.
Stems and bars depend on their construction, design, age, and use/abuse. Save 10 grams at your peril.