Mark the exact position on a sheet of rigid plastic where you want to drive a screw by scoring the surface with the point of the screw or with an awl.
Position the screw on the end of a magnetic bit and heat the tip of the screw with a lighter for about 5 seconds.
Drive the screw into the plastic while it is still hot. The plastic will melt rather than crack as you drive it.
Back the screw out of the hole if the end sticks out of the other side of the plastic. Drive the screw into a scrap piece of plastic of the same thickness, then grind off the tip with a file. Remove the screw and drive it back into the original hole. Its end will now be flush with the surface of the plastic.
Drill a pilot hole through rigid plastic whenever you want to drive a screw all the way through it to hold it to something else. Use a drill bit that is slightly larger in diameter than the shanks of the screws you are using to allow for thermal expansion of the plastic.
Use round- or oval-headed wood screws to screw rigid plastic and avoid sinking the heads by countersinking. A countersink, which is a hole deep enough for the head to sink to surface-level, weakens the plastic and may cause it to break.