The easiest way to check for signs of wear is to grab the chain at the very end of the rear sprocket and try to pull the chain away from it. A brand new chain doesn't move at all, and the general rule of thumb for movement is that if you can see half a tooth deep, it's time to buy a new chainring.
Furthermore, care must be taken that the drive chain does not "stretch". They have a little bit of wear on each roller, so the chain gets progressively longer/different pitch, i.e. more distance between each link due to wear. The sprockets don't wear out like this, and if the chain is too long, the load applied to the sprockets is only applied to the last few teeth, not the entire sprocket/chain engagement. This is why sprockets develop hook teeth as the chain ages.
If the chain is relatively new, it will still have the same pitch as it engages the sprockets, and pulling the chain away from the sprockets will result in little or no movement. If the chain gets tired, all this added play in its rollers and bushings will increase the chain pitch (the distance between the links) a little so much that pulling the chain can get significantly further away from the sprocket. If this method shows a difference of close to half a tooth, then it's time to buy a new chainring.
