Lee Hazlewood sings "First Street Blues" all in D major. But it is a common trick, not only by street singers, to start each stanza a semitone or a tone higher. A single dominant seventh chord does the modulating. Larger intervals are rare, since they are a waste of voice range (when accompanied; see below). More sophisticated performers play a little "modulating interlude" which may arrive somewhere quite different. Think of Cat Stevens's "Morning has broken", where the third verse goes back to the original C major (from D major), though we may be tricked into believing that it went another tone higher.

In all these cases you cannot say the song modulates, it's just the arranger or performer. With some Gospel songs, however, modulation between the stanzas is more or less intrinsic, think of "Amen".

Some choirs, when singing a capella, manage to sink by a semitone within a stanza or two, so letting them modulate a semitone higher keeps them from dropping off their voice ranges.

Songs modulating in their own melody are quite common as well. Particularly modulating to the fourth or fifth and back again occurs in traditional songs, dances, and marches. Since the middle of 19th century, modulating by thirds became more common in folkloristic music, but is sometimes frowned upon still today. The same applies to gender changes (except for the ordinary parallels, of course).

Tom Lehrer is quite a different genre, borrowing from Jazz and Pop music.