Thanks Jerry, for introducing this topic. I have always considered myself a solid but "non-flashy" guitar player. Now that I'm teaching guitar lessons, I've had to really think more about how I accompany songs, what are the most important elements, and why.
Most of my students are not interested in playing flashy solos, they just want to be able to sing songs that they love. So my role as a teacher is to help them do that effectively.
I have come to realize through having to take all this apart for my students that the #1 thing that you need to accompany a song is to keep the rhythm going. You can make all kinds of mistakes but if you can recover and keep the rhythm going, most people won't really notice or care about other little mistakes.
I would rather hear someone strum down once per beat with solid rhythm than try to do all kinds of fancy stuff and lose the rhythm. A simple strum is really all you need to sing a song. Once I get students doing that, we move on to bass notes and strums or bass notes and fingerpicking. But you can start really simply and still have a very effective accompaniment. I think it helps motivate people to keep going when they get the satisfaction of being able to do it right away.
When I arrange a song to sing myself, I'm thinking about what would best put the song across. Like Jerry, I tend to use a style that fits the style of the song-- a flatpicking style for country or old-timey kind of stuff, fingerstyle for bluesy things, vamping for jazzy things, and so on. But often whatever style I'm using to accompany I add a few small things to give some variety or something to listen to in between verses. Usually it's pretty simple, built on the melody of the song, and using bass notes and bass runs.
A lot of times I find I'm hearing something that I don't know how to play, which is what forces me to improve! And sometimes I take a song in one style and I suddenly find something new in it by using a very different accompaniment. I've recently been singing "Keep on the Sunny Side" in a very quiet, fingerpicking mode that gives a very different feel to the song.
Those are my rambly thoughts at the moment -- I'm interested to hear more from others!