The confusion comes with the fact that glyphs are specific to your class and some are even specific to your talent specialization -- and there are a lot of them. To take a look at your glyph options, hit "n" on the keyboard and select the Glyphs tab.

Do glyphs apply to all specs?

If you're applying a glyph to spell and it is unchanged or unmodified in anyway across specs it SHOULD carry over. The only time it shouldn't is if a spell is replaced by a talent and in any case, it should still apply to the base spell and be removed as long as the talent is taken.

Do glyphs work in Shadowlands?

Shadowlands Inscription Items

At present, there are no known Glyphs in Shadowlands.