Upon startingmy 01 Neon the fuse light appears on the dash and dings about 3 times and repeats and then goes off as if nothing is wrong. I have checked all fuses and can't find any blown. Everything electrical seems to work. Please help.

 

that fuse indicator is for the IOD fuse if stuff like your radio and inside lights work then it is not blown press down on the fuses inside the fuse panel.

 

[] Still no luck...I made sure all fuses were pushed in tight...both inside and in the fuse box under the hood. I did notice the dome light is out and the bulb is new. Can you tell me where the fuse would be located for the dome light? Inside or under the hood?

 

I found this info searching the internet. This applies to a 95 Neon. Perhaps it may be of some help to you:

The dome lamp receives power from the 10 amp micro fuse located in the Power Distribution Center (PDC) on the M1 circuit. This circuit is HOT at all times. The ground path for the lamp is provided in two different ways.
One way is through the door ajar switches. Circuit M2 connects to all the door ajar switches from the dome lamp. The switches are case-grounded to the body. When a door is opened, the plunger in the switch closes, completing a path to ground.
The second ground path is through the headlamp switch. Circuit M2 is spliced in with the door ajar switches. When the operator turns the headlamp switch to the dome lamp ON position, a ground path is provided through the switch on the Z1 circuit. This ground terminates at the instrument panel left center support.
The time delay relay is used to allow a time-ON function for the ignition switch lamp. Power for the relay is received on the M1 circuit from the 10 amp fuse located in the Power Distribution Center (PDC ).
When a door is opened, or the headlamp switch is moved to the dome lamp position, a ground path is provided for the relay on the M2 circuit. This energizes the relay, closing the contacts. When the relay contacts are closed, power is provided through the relay to the M50 circuit.
The M50 circuit supplies current to the ignition switch lamp in the steering column. Ground for the lamp is provided on the Z3 circuit. This circuit splices with the glove box lamp, the ash receiver lamp, and the PRNDL lamp. The Z3 circuit terminates at the instrument panel left center support.
So, I would check the time delay relay since it supplies power to the circuit. Also, check for a good ground at the instrument panel left center support.