I used to, but don't any more. I shredded some of it, and used some for mulch. These days, I have so much compost from my woodchip paths, I don't need any extra. I bundle the grass tops and put them at the curb on yard waste day.

I throw it on a mulch/compost pile. good stuff.

Yes I have always composted the grass prunings. I have a couple of cold compost piles going that are adjacent to the front and back gardens. All the garden debris debris gets thrown on these. I try to hide the piles behind some shrubbery or at the back of the gardens. Periodically I'll move the piles to a new spot, and turn the old spot into a garden bed.

In general, the clippings from large ornamental grasses are bulky and somewhat slow to decompose but that's okay because everything breaks down eventually. For the small grasses (and many other perennials), I use the smaller hedge clippers and clip them up into small pieces and let those fall into place and contribute to the mulch around the plants.