Most trampolines I see in people’s backyards are round, while most of the trampolines I see at the Olympics are rectangular? Why is this? Are round ones safer?
WAG, but if you were in the middle, the edges of a round trampoline will all be the same distance away, whereas a rectangular one wouldn’t. I suppose this might make a round one slightly more safe.
you don’t have as far to go to fall off.
a circular upper surround will be stronger than a rectangular one. if you use spotters without a surround they will be easier to do the spotting.
I would also WAG most usable bouncing surface area for material used - home units are more price sensitive items. Here for example is an Olympic trampoline for $1765. OTOH you get roughly the same bouncable area for under $500 going round. (And then add on the enclosures for both.)
It’s probably also easier to make the surface. The corners and sides of a rectangular trampoline are going to be under different sorts of stresses, and you have to design it to withstand whatever stresses are highest. If it’s circular, though, the stresses are going to be the same everywhere on the edge.
Ok, it makes sense for home one’s to be round, then why are Olympic ones rectangular?
Cost and safety are not the same limiters for Olympic ones; standardization and compliance with tradition are. In short, they felt that looks better.
The round trampolines tend to bounce you back to the center which is good for home trampolines. There are (were?) regular rectangular home trampolines. Some of my friends had rectangular ones while I had a circular one. I like ours much better for recreational jumping. The rectangular trampolines react differently depending on where you land and they seemed more dangerous and less fun. My WAG is that Olympic trampoline artists want the additional length rectangular trampolines offer to gain room in one direction for very high jumps and tricks.
No answer to OP directly, but just recalling that when I was a kid and played at lots of different homes/friends’ places, they were always rectangular shaped. I can recall at least 4 (5?) of them atm, and never a round one. My elementary schools all had rect ones also. They (round ones) certainly might’ve been elsewhere, not saying they weren’t. This was back from ~70’s thru early 80’s or so, fwiw. It wasn’t hard to get bounced to an edge where leg/arm went thru springs - OUCH! Doesn’t seem too happen as much with round ones (confirming Shagnasty here).
I definitely see more (all??) of them round ones now, with the ‘cages’, too (good on those cages being available now!). I flew/was bounced -off more trampolines than I should’ve, no doubt. Daughter was bouncing on round one earlier today, I think (at a friend’s). There is a ‘raised’ pool under it so when they get bouncing hard enough,water goes spraying everywhere - kinda fun looking 
When I was a youngster in the early sixties there was a business that sold bounce time on small rectangular trampolines set at ground level over concrete pits. It was outdoors, though there must have been a small office, and maybe a dozen trampolines. I remember going there several times and there were always others bouncing, but I think it went out of business fairly quickly.
Here is someone bouncing on a square trampoline in a backyard. (video)
And here, just because I’m a sick puppy, is the single most disgusting trampoline accident video you will ever see. I’m serious; if you watch the last seven seconds of this video, you will never be able to erase this horrifying image from your mind. You probably should just forget that I ever mentioned it, and move on to something more pleasant. Don’t click on this video.
That looks fake to me - his leg seems considerably longer than it should be, and changes as the camera changes hands. I could be wrong though, maybe I’m just looking for a way to tell myself that didn’t actually happen, because if it did, that’s horrifying.
There are no corners to land on for the inexperienced trampolinist, on a round trampoline. If you’ve reached Olympic standards, you shouldn’t need to be worrying about that.
You know, you may be on to something. It does look a little contrived. For the sake of our souls, let’s hope you’re right.
Round trampolines take up more usable space,which a gym may be short on. Also, it is certainly easier to spot someone from the side on a rectangular trampoline. I’ll grant you that hand spotting on a trampoline is difficult no matter what. I’ve usually seen belt spotting.
According to the wikipedia article on Trampolines, the inventor actually designed it with multiple people in mind. If you think of two people on a trampoline, then a rectangular shape makes more sense.
http://en.wikipedia.org/wiki/Trampoline#The_first_modern_trampolines
I also remember the diving coach using a trampoline for training as described in the article when I was in college in the seventies. There was actually quite a bit of overlap between the gymnastic, diving and cheerleaders.
I thought I was imagining that. I remember one just like that near my house in Massachusetts. It seems insanely dangerous now.
This video has been circulating around the net for a while, and it’s been generally regarded as fake. Pause and just look at the length of his “leg”.