I used to use a product similar to (or the same as) the one recommended by Sam. I think I once had an incident where the metal base got hot enough to damage the cup, so now, I always put water at the bottom of the cup, before I put in the oil. Anyway, the cups hold much more than enough oil to be lit for the requisite duration.
More recently, I've been using plain cotton wicks, like these. I have the little glass cups that fit into standard candle holders, with cylindrical metal wick-holders in the middle. The cups and wick-holders are of the form of those pictured here. It's a little messy to do the setup each night. I use a toothpick to poke out the old wick from the wick-holder, and then to push the new wick through it. In this manner, I can re-use the cups and wick-holders indefinitely.
I have not had any problems recently with either fire or unreliability. I think using glass cups specifically, and not plastic ones, is necessary to prevent fire issues. For reliability, I have two tips, both of which probably apply similarly to "genie-esque" lamps:
Make sure that your wick extends all the way from the very bottom of the wick-holder to a decent height (let's say about 1/4-inch) above the top of it. That way, it will draw up oil from the very bottom of the cup, will give you a large enough target for lighting from your shamash, and will produce a decent-sized flame. You can calibrate how large your flame will be (and therefore how quickly it consumes oil) by adjusting the length of wick coming out of the top.
Pre-saturate your wick with oil. I do this by simply setting the wicks up in the cups first, and then pouring the oil onto the tops of the wicks. When you do this, your flame will have plenty of fuel immediately when you light it; if you light a piece of wick without any oil in it, the string can burn away before it gets a chance to draw up any fuel, and you end up with a quick burn-out.
(Note that the product links here are for illustrative purposes only; I'm not endorsing the specific products.)