Ideally on a reusable tarp, one intended to last for years, like one suitable for camping I punch in extra grommets or sew in ties. Grommets, ideally the sort you get from a marine supply house intended for sails, heavier brass with spurs, get mounted at roughly 12" to 16" centers. Ties work better in the middle.

Construction site or utility tarps are much more temporary sorts of things. Hardware store grommets work pretty well if you run a strip of industrial grade duct tape on both sides before you hammer in the grommets.

I have also simply used duct tape to form loops at the edge. I run the tape down the tarp for about a foot, arrange for a three inch loop of excess and then back on the other side of the tarp for a foot. The tape top and bottom sandwich the tarp. These can be improved if you reinforce the loop by laying in a piece of fiberglass reinforced strapping tape or light line. The ends up on the tarp can be kept down by overlaying with a 3" piece of tape at right angles. The trick here is to round the end so there are no corners to lift.

I have had decent luck with a plastic gripped device that resembles 3" long plastic tongs that are held close with a sliding loop of plastic. Placed so the jaws ride up against the thin rope reinforcement at the edge of the tarp they don't slip. I have bought, over the last couple years, perhaps a dozen of them and they sell for $1 to $2 a pop.

They seem to work well enough. Fairly light but a bit bulky to use as a replacement for grommets and ties on hiking gear but two or four in a kit can be quite handy. I keep a set of six of them in my truck for hauling and have used them with utility tarps and plastic sheeting. On thinner plastic I lay on a piece of duct tape on both sides to spread the load if the attachment is going to be under strain. I have seen ones similar to these under different names.