Yeast-leavened breads (no matter whether they're white, whole wheat, or multigrain) are going to be your best bet. They have longer baking times, so a power outage will represent a smaller fraction of it, and the yeast keeps on working over time. So if the power goes out when the outside structure is set but it's not baked through yet, the inside won't just collapse. You'll still be a lot better off with a 15 minute power outage than an hour, but I think you could probably manage through longer ones too. The tricky thing will be telling when it's done - you obviously won't be able to trust a time from a recipe anymore, so you'll have to rely on testing. Depending on the specifics, you may also end up having to cover the top with foil or even reduce the temperature a bit to finish baking, to avoid over-browning it.
Quick breads - things like muffins and pound-cakes, soda-leavened with a dense crumb - will also fare okay. For example, I've had the power go out halfway through a 30-40 minute baking time on muffins, and just let them sit in the still-hot oven for the rest of the time plus an extra 5-10 minutes, and they came out okay. Not perfect, but I certainly wouldn't complain. If on the other hand you have a long power outage, especially early on in baking, you might be out of luck. If the leavening is spent before it cooks enough to set up some structure, it'll collapse on you.
If the outages are really short, on the order of 5 minutes, I wouldn't worry about much of anything at all as long as the baking time is longer. The oven already cycles power on and off to maintain temperature; with an outage that short it won't have time to drop before it gets a change to recover.
It also helps if your oven is good - better ovens hold their temperature better, releasing less heat to the surroundings, so it'll take longer when the power goes out for them to drop to a too-low temperature.
Finally, I would encourage you to experiment a bit. If you don't mind if 1 in 4 times something comes out a bit messed up but still quite edible, just go for it. And having things come out less than perfect will help give you an idea how much you can get away with, so you don't spend all your time avoiding things that you actually could have made!