In using the term compound, it may help to keep the distinction between the lexical classification of a word and its role in a sentence. The OP's example has a compound direct object
to sing and [to] play
A direct object has a function in a sentence (namely to take the action of the main verb), and it's composed of two of infinitives (which are words of the lexical class verb). An alternate expression could contain a compound main verb
He plays and sings
Compound verbs, on the other hand, are conjoined verbs that act together as a single verb with its own independent sense. These are rare in English. One response here has suggested
bow and scrape
but this is really a set phrase of two separate verbs, with scrape meaning to bow awkwardly by drawing the foot back across the ground.
Another suggestion is stirfry, but this really means stirred frying, with the first verb acting as a modifier to the first.
About the only pure compound verb I can find is go and do, meaning to deliberately perform an action, usually one frowned on:
He called the boss a jackass? Did he really go and do that?
Note that this verb is an idiomatic unit, different from
Did he really go to church and do an act of contrition?
Also note that both parts of the compound verb take tense:
Yes, he called the boss a jackass. He really went and did it.