No single method. Acronyms, splits, puns and double meanings.
Wordplay is the category for clues that do not fit the other categories. It covers acronyms, split words, puns, double meanings, names hidden inside ordinary phrases and anything else where the clue is playing a game with the word rather than defining, reversing or pronouncing it.
That looseness is deliberate. Rigid clue types produce rigid puzzles, and after a few weeks a solver who has learned the six categories can process most tiles mechanically. Wordplay is the section of the game that refuses to be processed, and it is where the writing gets to be actually funny rather than merely fair.
A lot of it comes from band names, which are the richest wordplay material in music. Acts have been naming themselves with puns, acronyms and creative misspellings for seventy years, and a name that is already a joke gives a clue writer something to build on. A clue about musical insects spelled creatively is not inventing a pun. It is pointing at one that has been sitting there since 1960.
The rest tends to come from surnames, which turn out to be full of buried ordinary words. Once you notice that a large share of surnames are occupations, patronymics or places, a whole category of clue opens up, and quite a few artists discover that their family name has been a puzzle all along.
Cryptune contains 74 wordplay clues, which is 4% of every clue in the game, spread across 70 puzzles. They break down as 50 on artist puzzles, 17 on album puzzles, 7 on song puzzles.
musical insects creatively spelled→BEATLES
From The Beatles. The clue does not have to invent anything. The pun is already in the name, and the clue simply describes the joke without naming it.
one who shears sheep→SHEERAN
From Ed Sheeran. A surname reinterpreted as an occupation that does not exist. The clue is grammatically about a person and phonetically about a name.
in perfect harmony→NSYNC
From NSYNC. A phrase meaning coordination, respelled as a stage name. Two transformations at once, which is why this sits in wordplay rather than soundalike.
rapid eye movement→REM
From R.E.M.. A clue that is a definition of the acronym rather than the band, which is exactly the kind of double life a good wordplay clue exploits.
If neither the synonym reading nor the opposite reading gets you anywhere, you are probably here. Wordplay is best treated as the fallback category, since by definition there is no single signal that identifies it.
Read the clue as a description of a word rather than a description of a thing. "One who shears sheep" describes a job, but the answer is a surname that sounds like that job. The clue is about the word, not the occupation.
Check whether the clue is spelling something out. Phrases where the initial letters do the work are common, especially for acts known by initials, and they are invisible until you look for them.
A wider sample, clue on the left and the word it hides on the right.
Back to the clue school, or see how clue types vary across the eras on the decade pages.