ANY TYPE OF NEW OR USED COOKING OIL CAN BE USED. I have tried oil candles with everything from BRAND NEW olive oil to the cheapest stuff I can find that was used to fry FISH and it's all the same. A USED COOKING OIL candle with a properly adjusted wick makes NO SMOKE and NO ODOR. There will be a wisp of smoke for 2 seconds after you blow the flame out that will have a faint odor of what was cooked in the oil, and if you push the wick into the oil to extinguish the flame there won't be any smoke at all. USED COOKING OIL is the only fuel I have found that will NOT stink up your house. There is the $40 / gallon ultra refined parrafin oil that promises less odor than normal parrafin lamp oil - I can't afford to stockpile that, but I can afford to buy and use and save cooking oil and reuse it for FREE.
One of the simplest ways is to use something shallow like the bottom of a pop can - you can even use one that's been crushed. Cotton string, candle wick, even a rolled up bit of paper towel will work as a simple wick. Use a matchstick to adjust it (push it around to have more or less of is hanging over the edge) so that it's bright but doesn't smoke. ANYTHING THAT USES LIQUID FUEL WILL MAKE SMOKE AND SOOT IF IT IS OVERFUELED. Because of its viscosity, USED COOKING OIL doesn't wick upwards very well so the device has to be very shallow. You will have to monitor it frequently or it will run out and extinguish itself. AS WITH ANY OPEN FLAME, CANDLES ARE DANGEROUS AND SHOULD BE USED ONLY WITH GREAT CARE.