To keep your broom from sliding off the wall or falling over, try giving it a hat. A hat? Sure! Head to your local dollar store (or dig under the sink for an old pair) and pick up a rubber glove or two. Chop off one of the fingers and slide it over the end of the broom.
It should hold tight and give it a little gripping power against the wall or corner it usually resides in. Since you'll have a few fingers left on your glove, try topping off your other mops and brooms and don't forget that one that lives down in your garage too! If you're not into dissecting a perfectly good pair of gloves you could also wrap a few rubberbands around the end if you have them!