In some sense we have already reached the singularity (or more precisely the A.I. has reached it). By some definitions the technological singularity happens when an artificial intelligence starts autonomously improve itself.
In a way thr already happens when the developers use A.I. coding tools to create better models. Of course that's not strictly autonomous (unless if the machine is playing us and using us to improve itself this way).
This week GitHub released Spark - their A.I. powered app generator. It can do simple apps from natural language description given to it. It's intended for small, simple apps at this point, but they don't really know the real capabilities of the generator. So in theory it could be hooked to itself and told to keep writing a better version of itself.
It might not be able to do that just yet, but neither did we humans be capable of doing what we can achieve today just a few years ago. And our learning capability is somewhat limited.
In any case, if this happens, it won't probably be the superintelligence we have been waiting for (or feared). It would just be a simple machine tasked to improve itself indefinitely. Of course it could end up doing so at the expense of using up all the energy available in the universe consuming everything else in the process.