How do I fix rolling stock that wobbles?

Depending on the car, I have used different methods.  One way, if the truck is removable, is to use a thin washer on the shaft to snug up the truck. When I say thin, I sometimes use a hole reinforcement that you use on paper in three-ring binders. I still have some of the old gummed ones and they work great. Another way is to wrap thin, black copper hobby wire around the shaft of the truck post to snug it up and simply twist it tight.   You can't see the black wire on the truck.  These methods take out the wobble caused by loose fitting trucks, and there are lots of them with this problem.  Just don't make this type of fix too tight that the truck doesn't turn properly.

If the trucks are riveted on, I fire up my press and cinch the rivets down to minimize the wobble.

The rivets used for decades to attach Lionel's plastic trucks - be they postwar, MPC or the more modern LTI period - have always been some what loose. The issue for me being an 027 guy, and also an operator who like you, likes some degree of realism, is when pushing a train backwards. Because of the loose mounting of the trucks, when going backwards through curves, the weight of the train, which is solely against the knuckle couplers, will cause the wheels of any particular car to rise up and derail on the curve.

If you need to see what I mean, take a piece of rolling stock with a rivet held truck, grab the knuckle coupler and see how much it moves up and down on the car body. If it moves a lot, you're likely to suffer derailments from that particular car. Also bear in mind, people tended to run shorter length trains during the postwar period. The longer the length of the train, the more likely you are to have derailments when pushing the train backwards into 027 curves and switches. This could also apply to 031 curves too.

So a standard procedure at the Brianel shops is to drill out the rivet and remount the trucks to the car body with a truss head screw and a lock nut. I use an 8-32 screw and nut, which has the advantage of fitting snugly within the way the plastic trucks are made, so I don't need to hold the nut with pliers.

Depending of the type of rolling stock, like with a gondola or flat car, I tend to use a pre-blackened hex or button type screw for appearance. Sometimes I use a 6-32 with these, and then you need to hold the nut with pliers as you are reattaching the truck. There have been times where I also go to the trouble of painting the screw the same color of the car body.

Now with drilling out the rivet, there is always the risk of melting the hole in plastic bodied cars, making it larger. I always use a fresh drill bit made for metal and a drill bit slightly larger than the inner diameter of the rivet. The key is to shred the outer lip or ring of the rivet: The largest diameter of the rivet that is on the bottom of the truck.

I've gotten very good at this and can do it with a bare minimum of melting to plastic. The key here is to angle the drill so that you are first getting that larger lip of the rivet shredded. And then also to not do the whole procedure in one step. It's a good idea to stop and wait for a moment to let the heat created by doing this, dissipate. 

When you reattach the truck, you first want to check for any flashing on top of the removed truck: I cut it away with a razor blade. Then you want to tighten it all the way, so the truck will not swivel to and fro. Then. slowly loosen the screw just enough so the truck freely swings to the left and right.

With postwar metal trucks that attach with a C-clip, Ron Arndt's suggestion works. Even more modern sprung die-cast trucks, factory attached via a screw can still be loose, so a very thin washer can help with those also.

Gunrunnerjohn's suggestion is also good, unless of course, you don't have the same equipment and tools that he has. Hence for that reason I started drilling out the rivets and have been doing it for so long, it's become second nature for me.

And it works for me. I haven't had a derailment from backing up a train through curves - even "S" curves - (other than my own operator error) in longer than I can remember.

And one more tip Coolhand: Since you are likely buying used rolling stock, you want to check the gauge of the fast angle wheel sets. The normal tolerances on them are pretty forgiving, but there is some variance between the different manufacturers. You want to be sure the gauge of the wheel sets on one given truck are the same... you can visually compare them. They can differ from truck to truck, even on the same car, but not on the same truck. If they are different on the same truck set, this car will be more prone to derail on switch tracks.

The two rail folks sometimes fasten the truck to the bolster with a screw and a coil spring. The coil spring keeps the truck from wobbling on the bolster.

You can also spin the wheels to see if an axle is bent causing the wheel to wobble.