In the same month I purchased the car (after putting about 1,000 miles on it), the head gasket blew - estimated cost of repair $800+. After doing online research, I discovered that it was a KNOWN issue with the 1996 - "new" Neons. The National Highway Traffic Safety Administration website had several postings from owners of these cars, stating they had to have the head gasket replaced MULTIPLE TIMES.
Called Daimler/Chrysler, and they wouldn't admit anything and blew me off. The car sat in front of my house (still making $220/month payments on it to the bank and keeping full insurance on it because it's financed at $100/month) for another 5 months without me driving it until I was able to locate a Dodge 5 Star dealership that only charged me $200 for the repair and billed Chrysler for the rest (in my opinion I shouldn't have had to pay a dime).
About a month later I had a bearing blow, and another gasket - I went BACK to the service center where they kept my car for 2 weeks and finally replaced the short block altogether (they didn't charge me because it was related to the head gasket blowing).
I drove the car for another 3 months and then my spark plugs fried (just figured this out last night) - I'm also getting clicking sounds around turns which is leading me towards worn CV joints. The car is now completely gutless compared to the first few weeks after I bought it. It is starting to shift extremely hard, it has major hesitation, and it's making a "THUNKING" sound on hard acceleration.
The trunk leaks like a sieve.
The fan is EXTREMELY loud.
A smell (not unlike CAT URINE) comes from the air conditioning (which is also mentioned at the NHTSA website).
Going through puddle the undercarriage gets wet and the steering ceases to function (I nearly ran into a cop because I couldn't steer - thank god the brakes actually worked).
In my opinion the car isn't worth more than $50.00 - and I'd get rid of it in a heartbeat if I could find someone to pay as much as I owe on it.
I've had the car for about a year now, and I've had more problems (primarily MAJOR) with it than with a beat up 1983 Honda Civic 1500 that I drove for 5 years - not to mention having it wrecked twice in that time. It still runs better than the NEON.