My question is simple. Do I need to oil a chain on a single speed bike?
Yes. You should follow the normal guidelines for oiling your chain. If it is squeaking, you know you've left it too long. Do the following every two or three weeks:
- Thoroughly clean the chain with some degreaser.
- Let it dry off. If the new lube comes into contact with degreaser, they won't be happy with one another, and the lubrication will not be as good.
- Apply one drop of chain lube to each bushing on your chain. Apply these on the bottom run of the chain, below the chain stay. Gravity and the centrifugal force in step four will help to work the lube into where it needs to be.
- Let the bike sit for a minute, then spin the pedals for 2 or three minutes. Be careful while spinning the pedals, especially on single-speed drive chains there is a real risk of injury from getting a finger stuck between the chain and the chainring.
- Clean as much lubricant off the chain as you can. It only needs to be inside the bushings. Any left on the outside of the chain will pick up debris and increase wear.
Although wear and tear does not affect a single speed drivetrain perfomance to the extent it does a geared drivetrain, a little maintenance will provide your single speed with an even longer lifespan.
Lubricating the chain helps ease the movement of the pins, rollers, and plates of the chain, and will therefore increase the life of your chain. Although this link is for a clock website, the images do a pretty good job of illustrating why lubricants are necessary in general. With no exceptions that immediately come to mind on a bicycle, anytime you have moving parts you are going to need some sort of lubrication. Gears or not, your chain should remain lubed.
As was pointed out by another answer, anytime you start to hear squeaks coming from your chain it's time to re-apply lube.