We err on quality over quantity, but I won't thumb my nose at lesser quality woods.
We only have one stove to feed, since our woodlot is small and my scrounging is limited to my time and wares (only a trailer, no pickup truck of my own) - and we already have access to a fair amount of ash, cherry, oak, and locust. That being said i don't have any problem burning anything so long as it splits relatively easy, dries quick and keeps well. Oak and Hickory are great when dry, but I'll take ash or cherry over them for to the relative ease of processing and the fact they can season well enough in six months and are ready to burn. I've burned Ailanthus, Catalpa, Sassafras, and Tulip Poplar in the stove and I'd do it again.
Because we only burn while home, the high quality, dense woods I often save for overnight burns, days when we're working outside a lot, or the really cold days. Otherwise middling hardwoods burn just fine only you have to tend the stove a little more often. The trade-off is that we don't get it too warm indoors. I've had to open windows in the winter because it got too hot inside while burning oak. Not complaining, but it has happened. Once the brickwork in our stove chimney heats up, maintaining a moderate fire with good coal bed is enough to keep it comfortable.