What's the best way to use my fireplace ashes for compost or fertilizer?

If you have any roses, then simply dump the ashes (make sure they're completely out and cold) around them and water them thoroughly. If you have a compost pile, dig them into the compost (don't just dump it on top and leave). If you don't have a compost pile or container then now is the time to start one ... they warm themselves up naturally (don't use any warm ashes, because they can start a fire) and will be perfect to fertilize everything in your gardens by spring. But do think about those roses, even if they seem dead or dieing, because roses need fertilizer especially right now, and the roses you get if you dump the ashes and water them in now will be well worth your effort.