Via Boing Boing, I saw this awesome article about what would happen if the Earth's rotation stopped. I don't mean all of a sudden like H.G. Wells once asked, i.e. everyone suddenly flying to the east at about 500 mph x the cosine of your latitude. I would link to the story but couldn't find it. But who cares, because here's the cool map:
I think you should take the color-coding as elevation only; there are good reasons to believe the middle of the landmass would have no green at all (keep reading.)
In essence, the maps reflect that without angular momentum, the ocean water would flow to the poles; right now it's 8 km deeper at the equator because of centripetal acceleration. But this isn't completely a thought experiment, because the Earth's rotation is slowing down, as a result of tidal forces (pay attention the next few New Years Eves and you'll notice at least one leap-second added.) In fact during the Devonian, there were about 400 days per year, which we know from fossil corals. The rate of the Earth's rotation will have fallen to roughly half its present value by the time the Sun goes red giant, although it probably will become tide-locked for a geologically brief period while the parent star expands. But nothing could survive on the liquefying cinder that will be Earth at that point, so we don't have to worry so much about that.
(If you really want to stop the rotation of the Earth like I personally tried to stop the San Andreas fault, we could all of us in the world fly to Belem, BR and then on the count of three start running due east to zero-out the Earth's angular momentum. Wouldn't work. In fact it wouldn't work even if all biomass in the world came along with us, because we'd have to go faster than the speed of light to do it. Besides not knowing exactly how the trees and plankton of the world will join us in our little escapade, by going around the world faster than the speed of light you risk going back in time like Superman as noted previously. Also of relevance, you can't. But I hear Belem is still nice for an Amazon port city.)
So besides the obvious map changes wrought by stopping our rotation, what else would happen? First and most obviously, a major climate shift. The oceans would be colder, because they're both at higher latitudes and deeper than our current oceans. This would considerably cool the overall climate of the Earth. If just the opening of Drake's passage was enough to put us into a sequence of glacial pulses, I would bet restricting all the world's water to the polar regions would put us into a very long-term snowball Earth phase. The land mass would be one continuous equator-girdling supercontinent with very little moderation by the oceans in the center (more on this later).
Of course as noted above we won't see the full effects of stopping the Earth's rotation but prior to the red giant age, there will still be some slowing. But then again the continents will have moved in the interim. Here's New Pangea, a mere 250 million years from now, 5% of the way to the red giant age (if you went back that far, you'd be at the start of the dinosaur age):
So for any future maps of the Earth that you smart geocomputer people make, don't just look at plate tectonics guesstimations, also look at the distribution of ocean water assuming a decreased (but not zero) rotation rate. (While you're at it, I want to buy property on Loihi ahead of the rush, i.e. before it breaks the surface of the Pacific. Work hard to find me a nice spot and in return I shall give you a shiny penny!) But look closely at the map - if the Earth's rotation stops, Loihi would be almost right on the coast! Also of note is that the wreck of the WWII dreadnaught Yamato would in fact be exposed on dry land.
The continuous belt of land around the equator highlights a second probable difference (and problem) with the no-spin world. The tropics drive evolution; biological innovation typically spreads from low latitudes to high latitudes. This has been shown to be historically true by an analysis of the fossil record, and it's true even when you look at the rate of evolution in current tropical ecoregions. The way the world works today, the equatorial regions are very wet, because of moisture from the oceans and east-west currents that drive moisture inland. But with no rotation, what would the polar ocean currents be doing, if they exist at all? If there are no north-south currents, then the center of Equatoria will make the Atacama Desert look positively lush. Not only will evolution slow as a result of the disappeared tropics, there will be less opportunity for biodiversity to appear: now there is only one continent, and all its climate zones are continuous east-to-west. That means there are no climactic gene-flow barriers. This is bad because if there's a problem in one part of the continent - a blight on critical grasses, an animal virus, an eruption that further cools the temperature at that latitude - there are no refuges.
(Take a minute to look back at that map of Future Pangea - it's also interesting to think that right now on Earth, we're in an odd period where the continents are near the point of maximum isolation from each other; we may have just passed it a few million years ago, right before South America joined North America. Coincidence that the planet's first intelligence appeared out of this era?)
The equatorial areas that were once abyssal planes will be undergoing a nice post-oceanic rebound, like much of Canada still is after the weight of the glaciers disappeared. For Canada this means all those awesome lakes and waterfalls, but if my other guesses for the climate of Equatoria are right, there won't be any water at all, except near the coasts, and it will likely be frozen. Maybe there will be two isolated ecoregions - two coastal tundras, separated from the Mars-like Equatorial Dry Valleys of the interior.
This of course neglects the most devastating effect of tide-locking: the sun-facing side would be cooked, and the far-facing side would be frozen solid. Even assuming some heat exchange between the two sides and without even calculating the heat of the sun-facing side, chance are the atmosphere would expand and all the water would be in vapor phase, and a lot of it would be lost to space. Even if somehow that didn't happen, you're still looking at two narrow temperate bands around the dawn/dusk rim of the Earth, with one piece of land at the equator of both. (For a long-dead discussion of terrestrial tide-locking see here.)
Long story short: don't stop the Earth's rotation. Like crossing the streams, it would be bad.