We don't really have a class: Like any writing, you just need to be aware of what's your goal. For a design doc, you want to say three things: (1) what are yo doing/planning to do, (2) why, and (3) what alternatives you've considered and why do you think they're worse.

We just use Google Docs, and it's easy to share the doc and have people add comments or propose changes. There's a template somewhere, but I don't like it much.