Stars get brighter when they're dying, not dimmer, so it's not likely to find a world that froze due to its star dying out. (If it was far enough away to survive the process, it was probably frozen to begin with.) Most ice worlds are simply far from their star in the first place, though not so far that their atmosphere freezes on, that's a different category. Water worlds usually just had a lot of water from the start. It's a common problem for superterran planets, due to their high MMWR. Superterran planets are unlikely to be dry, unless their oceans were boiling off in the past.
'Desert worlds' are a special case. The way the term is used is often confusing. A world isn't a desert just because it's dry, a true desert world still has an atmosphere and life despite being arid, otherwise it's just an ordinary barren planet. They're likely to be small and have thin atmospheres, since that makes it easier to lose water to space, without encountering a runaway greenhouse effect.
The above description doesn't apply to 'ice planets'... they're often literally made of ice, with rocky cores instead of iron ones. Ice planet with life on them will probably be rare, just like proper 'desert planets' are. Habitable planets are rare, in general. Ammonia planets are a fun type of 'ice planet', but complicated. Essentially, they need a stable red dwarf star and very specific conditions, but they might be able to generate life made of living high explosives. I'll have to work on that later.