Have any SHA-1 collisions happened? No SHA-1 collisions are actually known, though there are a number of cryptographic attacks to weaken it.

But, how likely is it that, of all the SHA-1 hashes computed since the invention of the algorithm, there has been a collision?

Not sure how to approach this, but it probably involves the Birthday Problem?

(if there's a better SE to ask this in, let me know)
In order for a collision to happen "by chance", about $2^{80}$ hashes need to have been computed. If 1000 hashes were computed every millisecond, then this would take 38,334,786,263 years. So, I doubt that this has happened by chance. 

Having said that, I would be very surprised if large government organizations do not already have SHA collisions (using better collision finding algorithms).