No need to say that there is a lot of variation in linguistic terminology. See Table I.I below from Cowie 1998: 5
Here are some most common ones - based on Granger and Paquot 2008.
- free combinations, e.g. to blow a trumpet
- restricted collocations (or collocations proper), e.g. to blow a fuse
- figurative idioms, e.g. to blow your own trumpet
- pure idioms, e.g. to blow the gaff
The last three categories are also known as "composites," as opposed to formulae (i.e. pragmatically autonomous utterances).
- semi-phrasemes (or collocations);
- quasi-phrasemes (or quasi-idioms);
- full phrasemes (or full idioms).
Granger and Paquot 2008
- lexical collocations (heavy rain);
- idioms (spill the beans);
- irreversible bi- and trinomials (bed and breakfast);
- similes (swear like a trooper);
- compounds (black hole);
- grammatical collocations (depend on);
- phrasal verbs (make out).
- complex prepositions (with respect to);
- complex conjunctions (as if);
- linking adverbials (in other words);
- textual sentence stems (the final point is).
- speech act formulae (take care!);
- attitudinal formulae (in fact);
- commonplaces (It's a small world.);
- proverbs (When in Rome);
- slogans (Make love, not war.)
The term "set phrase," when used in the broad sense, will include anything but free combinations (Cowie's classification above); semi-, quasi-, and full phrasemes (Melchuk's classification). For example, Mel'chuk 1998 says the following:
"People speak in set phrases, rather than in separate words, hence the crucial importance of set phrases. At the same time, set phrases, or PHRASEMES, represent one of the major difficulties in theoretical linguistics as well as in dictionary making" (Mel'chuk 1998: 24).
Burger, Dobrovol'skij, Kühn, and Norrick 2007 argue that "set phrase is a widely accepted term in English. [...] It thus includes all fixed expressions of a language, such as collocations, idioms, quasi-idioms, catch phrases, routines, proverbs" - in other words, it is equivalent to German Phraseologismsus in the broad sense (Burger 1998). To make things worse, some Anglo-American linguists use "idiom" or "collocation" in this sense, too. Here's what Burger et al. suggest:
To show how huge this terminological divide between Anglo-American linguists and the rest of the world is, in Granger and Meunier 2008 (23 papers, more than 400 pages), the term "set phrase" is used in two papers only (!), three or four occurrences only, the phraseological unit or phraseme being used much more frequently.
For more details, see Burger, Dobrovol'skij, Kühn, and Norrick 2007 or a collection of papers in Granger and Meunier 2008.