I've chopped off the ends of my fingers, I've removed several full fingers' worth of inadvertent hangnails, I've burned myself on every conceivable body part. I've forgotten that when mincing Scotch bonnet chiles, it's necessary to wash hands BEFORE urinating. I've set things on fire, I've caused boiling hot egg shrapnel to embed itself in my skin, and I've impaled my hand on skewers, artichoke leaves and cactus paddle spines.
1. During a kitchen fire, smother, don't wet.
Certain fires (cloth fires, for example) will be put out by water; if you've ever shaken a wet hand into a frying pan full of hot oil, though, you know what happens when water meets it; it spatters hot oil absolutely everywhere. Clamp a lid on a pot fire to starve it of air; salt or baking soda might work, but if the fire is truly out of control, use a fire extinguisher. You do own a fire extinguisher, right? A $50 insurance policy against a full-home fire?
2. A falling knife has no handle.
It happens to everyone; maybe the handle of the knife was jutting out over the lip of the counter, or maybe a sweeping motion caught more than food scraps. Once a knife starts to fall, however, get out of the way. Let it fall! The human impulse is to catch something that falls to avoid damage, but the knife will be damaged less by hitting the ground than you will be as you slice into the pad of your palm.
3. Work clean and work dry.
Clean as you go, and more importantly, dry as you go. Knives slip more easily on slick surfaces; greasy hands slide down hot pot handles more easily, and liquid of any kind spattered onto oil will cause popping that can cause serious burns. Don't try to chop in a crowded space; clean first, then wash your hands and chop in your newly-found room. Prepare everything beforehand (the much-vaunted mise en place) so you're not chopping raw chicken while you need to be tossing onions into a pan. You'll be safer and you won't be cross-contaminating things hither and yon.
4. Always, always away.
Always lay foods into a pan away from you. This way, if grease spatters, it hits the stove or another pot, rather than your delicate skin. Always lift the lid off a pot far edge first so you don't get a steam burn. Always turn items you're sautéing over away from you so you don't splash grease onto your hands, and always, always sprinkle salt, herbs, etc. away from your body so they don't land on your floor and make you slip. (Professional kitchens have mats for just that reason; your kitchen probably doesn't.)
5. Pick up the knife and look at nothing but the knife.