What are the advantages of a liquid-filled compass over a dry one, and vice-versa? Also, I have seen several posts that refer to a button compass. Where do you get a button compass?

The idea behind liquid filled compasses is to dampen the needle oscillations, which is a good thing- other devices were developed, but it still took some fiddling to get a good bearing even on terra firma, and it was almost impossible on a boat in choppy water, for instance. A side benefit is that it makes a diving compass much easier to engineer, since the liquid inside won't compress, the compass housing itself doesn't have to be strong enough to withstand the pressure. Without the liquid, you'd have to have a pretty massive housing.

The biggest disadvantage is "bubble trouble", which happens to a LOT of liquid-filled compasses over time, even some expensive ones... and in general, the smaller they are the more likely they seem to develop bubbles. Bubbles are annoying at the very least, they can affect accuracy if they get too large, or they get under a compass disk, and you never really know if they're going to keep getting bigger or not. I've also seen the liquid deteriorate the paint on the needle over time.

In previous wars combatants and spies were issued various compasses disguised as buttons, or concealed within hollow buttons (to aid escape from captivity), and perhaps that's where they got the name, but often "button compass" refers to any compass of about that size and shape. I've seen them called that more on this site than anywhere else.

This happened to my watchstrap compass that I had ..... suprisingly, on my watchstrap. After about a year of continuous wearing I couldn't read it at all because the liquid was so cloudy.