A rappel is no panacea for poor scouting. as a rule of thumb, don't drop into something if you can't see the whole descent. A 10-foot drop might prove impossible to climb back out of, and there might be a 50-foot drop below that, leaving you stranded in between. Better to spend the night hiking out—and staying warm—than trapped in limbo on a cold ledge.
That said, for a cheap insurance policy, carry a 100-foot length of 7mm accessory cord purchased from a climbing shop. Cheap, compact, and disposable if need be, 7mm cord is rated for over 1,000 pounds. Doubled over, it will give you 50 feet of freedom, or 100 feet if tied off and left. Coil it tightly and carry it in a small stuff sack to keep pack tangles at bay.
Trees or big rocks make the best anchors and should be tied off with a Bowline knot (realknots.com). If no anchors are handy, try burying a ski pole, branch, or clothing packed with snow. In hard snow, dig a deep, horseshoe-shaped trench and carefully wrap the rope around the snow in the middle. If you get desperate to extend your reach, try scavenging full-strength webbing off of your pack or boots to tie around anchors.
When it's time to rappel, wrap the rope around a body part (forearm, thigh) a few times to create extra friction. Always wear your gloves when lowering to prevent rope burns. Avoid the temptation to throw your gear downslope—you never know when you might need a crucial item.
And don't panic.