I'm a fan of folding bicycles. The need to fold adds an extra design constraint and I like seeing the clever designs that people come up with to work around the limitations. The only kind of folding bike I do not like is when they saw the main frame in half, weld a giant hinge in the middle and call it done. There are advantages (drivetrain stays in place) but I think it's lazy and causes the creaky/flexy frames people associate with folding bikes.
My favorite folding design is the Reise & Mueller Birdy and I have one. There's a hinge at the bottom of the front fork and another for the rear triangle at the bottom bracket. Both wheels fold under the frame, the seat post drops to lock in the folded rear triangle, and the stem folds down to the side so the whole bike locks in place around the size of a large suitcase and you can use the saddle as a handle. Here's what my bike looks like half folded, which I use instead of a kickstand:
What I really like about the design is that both hinges are used to provide full suspension for the bike. The front fork is a fairly unusual leading link (also used in some motorcylces) and the rear triangle is a fairly typical traveling rear triangle used in mountain bikes and Bromptons. The absorption is provided mainly by elastomeres, which aren't that high performance but they work well enough to allow me to ride on cobblestones comfortably and help keep the smaller wheels on the ground.
I've never taken a picture of my bike fully unfolded but if you want a better look, here's a video of an identical bike except for the tires:
The main downside to the Birdy is the price. This particular model sells for over $2k, which is a significant premium for an aluminum frame and mid level components but roughly in line with other premium folders. It's possible to save ~$500 by going with the original oversized tube farme design but I like the fancy monocoque version. As with most fully suspended bikes, if you stomp on the pedals the frame will pogo. I angle my push forward a bit and don't have a problem but it was something to get used to. Finally, in terms of negatives with the fold design, having a traveling rear fork means the chain goes slack when folded under. To fix that, there's a chain tensioner attached near the bottom bracket that sticks out just past the chainring.
All in all, I think it's a great city bike. It fits in the bottom of my closet, I can take it on the subway if it rains, and I never have to leave it outside since I can stash it in a bag and buildings don't care. This last point was one of the major motivations for buying it since I wanted a nice bike and did not want it to be stolen.
In terms of speed, I do laps around Central Park in about 20 minutes or ~18 mph and can get it up into the low 20s mph to keep up with traffic–speed limit in Manhattan is 25mph–and hit the green light waves down the avenues. I have two limits around going faster. One is aerodynamics, my body position on it is basically like a road bike on the hoods. The other is that the gearing is 1x and set up more for hills so 100 rpm tops out at ~22mph. Note that I'm not especially fit and these are pretty high intensity for me. A long ride through the city is 5 miles and park loops are 6/12 miles.
So I was planning on that being my only bike until I moved but last week on Craigslist:
A folding recumbent! This is a 2003 Bike Friday SatRDay, which is a fairly rare bike because they were expensive and just didn't sell that many of them. Bike Friday's thing is that their bikes disassemble to fit in a suitcase and this one does that but it also folds up a bit which helps in getting it through doors and fitting out of the way in a corner.
I just completed the purchase on Wednesday night in time for the snow yesterday so the above photo is from my first short (about 1.5 mile) ride on it. It's a combo derailleur+hub and learned that the hub was missing the shifting pin. I also had a couple people try it out on the river walkway, which was fun. The plan is to use it for longer rides (e.g. 5 boro tour) and cruising around the park, it's a lot less practical for city riding than the Birdy. The rolling resistance is also noticeably higher, so that's something I'll need to fix up.