The Caltech lab instructions that I found while searching for the Maxwell Top describes a way of doing it using a torsional balance, so I decided to give it a try and see how difficult it was. To my surprise, it worked better than I expected so I went ahead and measured a few tops.
I replaced the torsion rod of the lab notes for a piece of fishing line (450 mm of 20 lb Test mono-filament). All you need is the line, a place to attach it and a stopwatch. Nothing more. This is my setup:

The measurement is very simple: you rotate the top in one direction, so the line twists, and let go. The square of the time it takes for the top to twist back and forth is proportional to the rotational inertia.
If you want to take absolute measurements you just need to calibrate it using an object with a known rotational inertia, such as a solid rod (what I did).
I measured Spintastics tops because the hole on the crown made attaching the fish line easy. These were the results:
Spintop Moment Inertia
QS full metal6.20
QS with TG cap4.31
HP with bulldog cap3.61
TG with ring3.26
TG no ring2.99
QS cap2.49
Bulldog cap0.53
The units are kg m2 x 10-5.