My example of a beginning is Ok Go's famous Here it Goes Again video. It starts off with a very fast regular beat, then switches back and forth between regular and irregular with both the voice, guitar, and drums starting at about 11 seconds. In the regular beat it starts out with, it slowly increases in volume creating a crescendo that leads into the song. This engages the listener and draws them into the rhythm and volume of the song.
One of my favorite 'popular' songs is Tom Lehrer's Oedipus Rex (Yes it's old, but it's very similar to popular music as it was meant to parody movie title songs). The song starts at 1:57, yes there is a lot of talking, and goes to about 2:21. At 2:31, the first verse comes in and lasts until 2:40 where the second verse comes in. At 2:52, the bridge to the next verse comes in and at 3:02 it transitions to the last verse which ends at 3:10 and immediately goes into the conclusion, there is no chorus, which is a little strange.