2002 Chevy Cavalier Sometimes Won't Start

So very recently when I try to turn on my car it will be dead silent, but the lights and
radio will come on.  The car won't itself start.  I try a few more times and eventually it
turns over.  For a few days it will be okay, but then sometimes it will give me trouble
again.

Is it the ignition switch that's wrong or could it be something else (like the starter
solenoid)?  I've had this car for a few years and it's quite beat up, so I am
contemplating whether it is worth the repairs or not.