What is the easiest way to light up a train wall display case?

PRRHSC,

    For me it was always good LED Track lighting that made a display case so easily seen,  You can use it from over head or the side wall, or even both.

PCRR/Dave

easiest way is  with  a  free flashlight from harbor freight

Train Room lighting.  Click on the link.

Easiest?  with some form of small spot light from above and outside the case.    to light each shelf inside the case, I would use christmas rope lighting.  They do sell rope lighting for various voltages, but this time of year you may be able to get the standard stuff at the big box stores on clearance.  I picked up several strings last year for like $8 for a 10 foot rope of LED lighting.  You may do better buying a 100' roll or some such.  Rope lighting can be cut at specific lengths, usually about every 12 inches, and is usually marked as to where it can be cut.  I like to use zip ties with the screw mount hole to attach rope light, but there are plenty of more expensive options out there in the christmas lighting sites.  

I'd probably consider LED strips inside the case, that would give lighting on the shelves without external fixtures.

the museum i used to make up displays for had a number of these 4' x 4' x 4" display cabinets...

the case was lit by a single florescent bulb with a UV shield and heat was never a problem.  the case has sliding glass doors not shown in this picture.  tempered glass shelves are key, though, allowing the single lamp to light the entire cabinet.

in your case with the vertical shelf spacing being tight, you might want to consider ditching the track.

cheers...gary