I think the best thing going (at least on the inexpensive end) are LED lights. They typically stay pretty steady so it shouldn't cause any flickering in your video.
When I first set up my studio at home, I had a flicker problem which was caused by the AC power fluctuating by a couple of volts. I had been through all the settings on my DSLR that can use flicker in the camera, but this was the power in the wall going up and down as factories a couple of blocks away turned machinery on and off. So any light that would respond to a dimmer switch, like my halogens, was slowly getting brighter or dimmer as the voltage changed. I used a multimeter to check this, and captured frames in Dragonframe with the meter in view, so I could see the voltage and brightness went together. (I switched the 1200 watt oven in my studio on and off to make the power change for my test, rather than wait for it to happen.) I had mostly halogens, and LEDs hadn't come in to general use yet, so I ended up spending a lot on a voltage regulator (a Double Conversion UPS) that smoothes the voltage out.
So a light that does not respond to a dimmer would be worth a try. Some fluorescent lights seem to work, they stay the same unless the voltage drops a lot. Others seem to drop a tiny bit, enough to show upon camera. I am able to use some of my fluoro tubes (which don't go through my voltage regulator, they are the regular ceiling lights) for soft ambient lighting, but not others. (fluorescent lights have a rapid flicker of their own, so you need a slow exposure, like 1/2 sec or longer, to smooth that out.
LEDs that don't dim with a regular dimmer would certainly be worth trying. LEDs are pretty much the up to date lighting choice these days anyway. I have been running a couple of them through my system along with halogens, so there was no flicker - I would need to test them on unfiltered power to see if they have a steady brightness on their own or not.
Thanks for the replys, I start looking at some LEDs.
LEDs are definitely the way to go. I have had problems with halogens causing flicker.
It is possible to use dimmable LEDs, but you have to watch out. The dimmers are Pulse Width Modulation dimmers, and they operate by switching the lights on and off, too fast for the eye to notice except to perceive less overall light. Unfortunately this can be a bit too slow for camera shutters. The way around this is to use 10k PWM dimmers. I have made up some light panels using strip LEDs and these dimmers, and have had no problems with flicker.
Also look out for high CRI LEDs. Colour Rendering Index goes downwards from perfection (100), which is what the human eye can see. A lot of cheap LEDs are about 60 or 70, and can give a colour cast on video. It is possible to buy high CRI LEDs (95+) and filmmakers are raving about them. There is a Crompton GU10 LED with 90 CRI that has a 15° beam, which might be very handy for a small spotlight. I am just about to buy some...