If in doubt, chuck more deeply or use a collet. A turning that comes adrift can damage both the lathe and you. A workpiece that shifts can damage both the mill and you. Stuff gets hot when cut. Let it cool before picking it up.
Metal cutting generally leaves a sharp burr.
Break the edges with a file or a de-burring tool before you release the work for general handling. Don't let kids, wives, husbands, girlfriends, or boyfriends close to metalworking operations without training or close supervision. Chips are extremely sharp. Long, curled chips from lathe turnings are especially dangerous because kids, wives, husbands, girlfriends, or boyfriends see only how pretty they are. You can get a very deep cut by handling such chips with your hands.