Turn on your hot water tap only and let the water run for at least a minute. Test several different taps -- the shower, the kitchen sink and so on. If the water remains cold, your hot water heater is malfunctioning.
Smell the water coming from your taps. A rotten egg smell can indicate a problem with the aluminum rod in gas hot water heaters.
Run a hot water tap full blast and observe the water flow for signs of weak pressure or spurts. This can indicate a clogging problem in your hot water heater.
Listen to your hot water heater for at least several minutes. Listen for popping, banging or cracking sounds. These could be signs of hard mineral sediments, which can eventually form chunks that will break your hot water heater.
Look and feel around the hot water heater, the nearby walls and the surrounding floor. Signs of obvious leaks, puddling, mold or a spongy floor are signs of a leaking and broken hot water heater.