"Sing" might be the most common and general one. Chirps and tweets are sounds of very short duration. Warbling involves alternating between several different pitches, whereas a trill is rapid alternation between only two pitches.

I would say chirp, sing, or tweet are all common with 'sing' being the most common I should think, but like John V Hall explained, there are tonal differences between each word. As for the songs of the cicadas or grasshoppers, 'sing'.