While we’re glad to have you in the Fiat family, we do regret to hear of the concern you’re experiencing. As our team is not technically trained we rely on the dealer to be our eyes and ears in the field. If you do end up connecting with your dealer to diagnose your recent jitter concern, our team is just a private message way and happy to provide an additional layer of assistance throughout that process if needed.
That is an odd one, ROC... but a great project idea. If you had said you felt it in acceleration in 1 & 2, I might suspect the traction control having a problem, or maybe motor mounts, etc. (You could try turning off traction control, press the ESC button, but I doubt it'll change the jitter.) I will assume the jitter is the rear, not something transmitted by the front.
I would start by putting the car up on stands and inspecting the rear axle. As you know, it's a pretty simple design and well tested. My first suspicion would be wheel bearings. They come as a hub assembly and can occasionally fail early on your model. Give the tires a tug to check if they wobble and they should rotate easily and smoothly. While there, check the tires for even wear and no cupping. Give them a spin and look to see the tread doesn't "wobble"... a slipped belt. Depending on your skill, the next step might be for a specialist. Inspect the shocks, bushings and bolts for proper function. If you replace the hub assemblies, buy a good quality part, factory, MOOG or similar. Aftermarket are half the price but can fail quickly... some under 20K miles.
If that is not the problem, a 4 wheel alignment might be in store. These can be inspected at home (see youtube) but is usually best done by a professional. While there, you could have the suspension and tires inspected, rotated, and/or balanced. IMO, most cars leave the factory slightly out of alignment and my 2018 500 was no exception. It's something most of us assume is OK, but in the last 50+ years, we've not had a single new car that was w/in specs on all 4 wheels. For the rear, most shops, including most dealers, are going to tell you the rear axle is not adjustable. Mopar (and other sources) sell a shim kit that allows the rear axle to be aligned. I had my dealer align my '18, but they didn't know about the shim kit and couldn't find the part number, so you might want to search the archives here to find the part number. (That's not uncommon. I had the same problem with my Ford Ranger.)
I hope that helps. Alignment doesn't usually cause a jitter, but your tires should last longer.
Thanks for the great advice Map. I'll take work on them in order and see what solves the problem. I'm suspicious and concerned that a motor mount might be the issue. It is most obvious when traveling above 40 mph in 3 -5 gears when accelerating. Goes back to rolling/ feeling smooth when letting off the gas. That's why I don't think it's tire or alignment related. I'm going to get it on a lift Saturday.
I owned a 2012 Fiat sport for 8 1/2 years. Take a close look, at your tire wear, on the outside edge of the rear tire. The edge of tread might look feathered up. You can feel that roughness when rolling. As map suggest, there is a Shim kit that will correct this situation with new tires.
If you had a bad rear wheel bearing, you would hear a wining noise. Your symptoms would be at any speed. I’ve replaced my rear wheel bearing, both sides went bad.
This has nothing to do with the rear end. The front half shafts, are known to fail on the 500. You didn’t mention, how many miles, on your vehicle.
This is what a failed motor mount, looks like on a Fiat 500. There are 3 of them. This one is located, under the battery box.
Thanks for the feedback Smark. Tires are wearing evenly and rolls quiet. My bet is on the motor mounts since the car is only jittery when trying to accelerate. When I let off the gas it rolls really smooth and tracks straight. I forgot to mention at the start of this thread that the car has 68,000 miles.