Fold up bikes are a great concept. There is one 'but' though, they are not nearly as robust as a regular bike and hopping a curbstone or a pothole can have serious consequences.
The pedals are usually a little lower over the road surface than on regular bikes and the wheels are small. They are also less rigid.
When I was a kid I would ride a fold up bike to school every day, for two years without incident, then one day one of the hinges suddenly gave way. It's very surprising to find yourself on two half bikes in the middle of a busy city intersection!
I had a Strida in London. It seemed like one of the best-designed folding bikes around: kevlar belt (instead of a greasy chain), folding and unfolding an A-frame bike took literally 10 seconds.
But here's the catch: the tires are all really small, so it's much trickier to balance, and it's much less comfortable than a full-size bike.
S&S bikes have been mentioned, but they aren't really folders they're simply bikes that are relatively easy to disassemble for travel whilst still offering a full sized bicycle and the stiffness and qualities that gives.
There are some small bikes along the Japanese "minivelo" style that require only minimal disassembly. Such as the Moulton TSR which is a fantastic bike with a ride quality not far from a full bike.
I would steer clear of the BSOs (Bike Shaped Object : http://www.outrider.org.uk/wiki/Bike_Shaped_Object ) that offer a folding bike for far too cheap a price. There's a reason those things ride like hell and don't have a good reputation.
I second KWD's recommendation of BikeFriday bikes. I have a Tikit folding bike. Its far more stable and durable than any other compact folder I've ridden and have taken it over curbs and through pot-holes without a problem. Basically the folding mechanism can't engage as long as there's weight on the peddles.
You will have some flex in the frame, but less in my opinion than other folding bikes I've ridden. All the issues with a lower slung frame still apply.
The killer features for me are: 1) I can fold it in about 5 seconds; 2) I can wheel it around while folded; 3) other than the frame it uses standard bike parts.
It all depends on how much folding you want. If you just want to be able to travel with it a bit more easily than a full size bike, you can do something like this:
S&S couplers are necessary for bike travel these days. Bikes now incur a $175 surcharge on any domestic flight I've been on in the past year. If you can get the bike into a suitcase you avoid the charge.
However, couplers on a real bike are nothing like a strida, brompton or bike friday tikit. These bikes are designed to fold up and unfold very quickly. They are also much smaller and can fit on public transportation where you're not allowed to take a full sized bike.
I used to cycle a lot when I lived in California. I knew several people that used a Bike Friday, and they were as durable and functional as a regular bike. I'd even see them on Century (100 mile) rides. http://www.bikefriday.com/
Why are they a great concept? Non-folding bikes aren't particularly heavy or unwieldy. Finding a place to park/lock your bike isn't hard in most cities (I think).
I guess I don't understand what problem a folding bike solves.
Probably in cities where crime rates are high. I've gotten 3 bikes stolen in SF already. The last 2 bikes used some heavy duty U-locks that almost cost as much as the bikes themselves. Some office buildings in SF don't let you carry a bike inside, but they don't care if you fold it up and stick it in a bag. Also, if you have a car, it's a lot easier to throw the folded bike in the trunk than taking apart the front wheel of a normal bike and squeezing it in the back.
In Amsterdam a friend of mine observed that the combined weight of bicycle and a lock strong enough to ensure that it will not be stolen is a constant.
We still haven't figured out the depth of this observation, does he mean that the lock is so heavy that the weight of the bike is negligible or does he mean that as the weight of the bike goes down the weight of the lock must go up...
I wonder how much he spent making the prototype.