What is the regularity of appearance of /uː/ and /u/ (or /ʊ/ in RP)? How can I be most sure deducing from spelling alone, that, say, "ooze" is pronounced /uːz/ and "wool" as /wul/? I know that English vowels are peculiar, but I don't want to look up the pronunciation of words in the dictionary that often.
Unfortunately, the occurrence of /u/ vs /ʊ/ is a little arbitrary, in particular because the /u/ vowel occurs in words coming from a number of sources. There are even a few words where either vowel is possible (e.g. "room", "broom", "toothpick"), though /u/ is possibly becoming more predominant in these cases.
However, here are some rules of thumb:
There is also a little idiolectal variation as I mentioned, and in Scotland, the two vowels are often neutralised by some speakers, so e.g. "full"/"fool", "pull"/"pool" are pronounced with the same vowel.
P.S. You can generally assume that /u/ and /u:/ are basically the "same vowel". Like vowels in general, /u/ will be lengthened before a syllable-final voiced consonant, so e.g. in "use" the noun [jus] it will be shorter than in "use" the verb [ju:z]. But this is essentially the same phenomenon as in e.g. "piece" [pis] vs "peas" [pi:z], or "sent" [sɛnt] vs "send" [sɛ:nd] etc.