I live in an old house, and along the steps leading down to the basement is a light bulb socket. It used to work once, but now even with a new bulb which works in other sockets, there is no light when I flick on the switch. Help in solving this mystery would be greatly appreciated.

1-Start at the breakers/fuses to verify that none are tripped/blown. Yeh, it's the most obvious :D but also the easist to check. Do other devices on this circuit still work?

The working back method:

1-check the socket voltage/condition- Check for voltage AT the wires connected to the socket, not where the bulb screws in. If the socket is defective or has loose, corroded or broken terminals, the bulb won't light even though voltage is supplied. If you're getting voltage at the wires, figure a bad socket. No voltage at wires, bad switch or wiring.

2-loose connections/bad switch-Make sure the wire screw terminals are tight. If back wired, check that none of the wire popped out-if back wired move wires to the screws. Test for voltage on both sides of the switch. No voltage at supply side, bad/broken wire or loose connection/splice elsewhere in the circuit. No voltage on load side with switch on (supply ok) bad switch.

4-bad wiring-You say the house is old. Depending on age the actual wire may be a problem.

NOw, the throw parts at it method;

1-replace the switch with a known good one. If it works problem solved. If not
2-next replace the socket. If that doesn't work someone will have to trace the problem.