Umlauts are one of the more consistent schemes at least, and generalizes decently well if you treat it as the indication that vowel is being centralized (i.e. applying it to "e" and "i" should yield unrounded central or back vowels - which is indeed the case in some languages; e.g. "ë" in Albanian is basically the schwa). If you map it to the frequency-based 2D vowel chart, you get something like this:

  i ü ï u

  e ö ë o

    ä a

Throw in another diacritic for lowering that can be combined (say, ogonek), and you get 20 vowel characters to play with - that's sufficient for just about any language I can think of, and with some care in mapping, also translates naturally across languages.

Most other systems are very haphazard largely because all of this stuff evolved as scribal abbreviations. (So did umlauts, of course; it just happens to be the lucky case where it ended up mostly consistent.)