First off, I think with modern high quality fabric it probably doesn't matter much. However, I almost always use a french double fold cut on bias binding. For years I used straight of grain but decided I flouted enough of the quilt police rules and could learn to do a proper "competition" binding with good mitered edges etc. some 10-20 or so years ago. But I typically make mine quite a bit wider than 1/4" finished as I am generally making some sort of statement by my fabric choice -- for example I love using stripes in my binding and cut on bias I love the way they zip around the top and I want them to show. I'm prolific enough that I'm pretty fast on making the bias and don't need to refresh myself on directions.
One thing I've learned from others on this board is I no longer press the bindings in half the long way. I just fold it as I put it on and am very happy with the results. I will typically make my bindings at the starting/cutting phase of the project, that is before I've pieced any of the block parts. I grab a piece of cardboard and wrap it around until I'm ready --- 18" long often bends but is a quick and easy way to count how many yards I have, one round being a yard and it still fits in my fabric bins.
It is really all on a project by project basis -- there are times when I just want a small narrow binding that disappears/the same fabric as the borders. Or, the piece of fabric I have might lend itself better to long grain or cross grain.
I feel that since I wash my quilts rather frequently that experience has shown the bias holds up better and the fold and hard edge of straight grain ends up breaking up sooner. I've had to rebind some quilts... for that I typically use just a single layer of fabric to help hold things together and may do some decorative stitching on/through the binding to help hold it together. For quilts that get washed less maybe not so much an issue.