If the fuse was "blown" then that would indicate a short circuit. If the fuse wasn't blown that would indicate an open circuit.
The most likely cause in either event is a broken wire. The 2nd most likely reason is the switch on the column that engages when the lever is in the reverse position.
I would start by trying to trace the wire starting at the bulb and working forward. You'll likely find the wire is broken very near where it goes into the mount.
Very possible ... best to check the wiring with a voltmeter checking for +12vdc.
the reverse switch in the transmission
Defective reverse light switch.
Where is the fuse located for the reverse lights
Check your fuses
There are a couple of reasons why a 2005 Saturn Ion's reverse lights will not work. The fuses may be blown or the light bulbs need to be replaced. Fuses and bulbs can be purchased at any auto parts store.
Both a stuck reverse switch and shorted wiring can be causes for reverse lights to stay on. Under normal circumstances, the reverse lights turn off as soon as the vehicle is shifted out of reverse.
The reverse light switch is faulty!! If you plan as a DIYfer be sure you have enough knowledge about automotive electricity!! Avoid fires!! Read and understand instructions!! You have only one life!!
Try bypassing the reverse light switch that plugs into the transmission. If your reverse lights come on, voila! Problem solved. Buy a new switch.
it's your reverse back up switch need's to be replaced that's located on the front of the transmission under the air intake piping
Try replacing the headlight switch. Do the front parklights work and not the rear ones? If so check the harness to the rear of the vehicle for breakage.
The problem is the reverse light switch. It is located on the top of the transmission that should sloved your problem.