I've got a 1990 MR2 SW20, recently got it Rego'd and road worthy after buying it.

The issue with the headlights is that they pop up, but they won't turn on. After driving for ~20 minutes and going over some bumps they have started working but turning the car off for more than 20 seconds makes them cease to work any more.

Going to try and check the grounds in the morning.

Do the lights come on with the high beam flasher? (pull stalk towards you)
If they do, the problem may be with the steering column dipper switch. (Not the part of the switch that turns the lights on and off)
The copper contacts can burn and the lights won't work or may work intermittently - a common problem.
Check out:
http://www.mr2australia.com/mr2play/tm.aspx?m=66156