Well we can easily talk in broad strokes what we know “the singularity” brings.
The main thing is that some point during this period of rapid change, commercial forces or whatever driving intelligence is behind it will order the robotics fabs to build a very large number of automated research and development nodes.
These would be automated research systems that explore, systematically and from base principles, every field of science. Biology, chemistry, physics, mathematics, all of it.
They would build up a fresh model of the world, with unbiased, robotically obtained data, from the beginning. And not just a model - at the upper ends of the process, optimized versions of every piece of machinery this civilization is interested in would be developed, from jet engines to nano robotic assemblers.
That is, at a certain point, my prediction is that in a relatively short period of time (probably 10 years or less), this civilization would have systematically discovered every principle of physics that can be found with a systematic search and exploited in our universe, and developed the machinery to within ~1% of the hard limit allowed by physics.
So they’d have nanomachinery that’s 99% as good as the best possible, rocket engines that are 99% as good, and so on and so forth. The singularity is an S curve.
As for what sort of things we know that you could do with such technology:
a. You could play with biology, including human biology, like legos. If humans are still alive at this point, and these humans have some remaining autonomy and purchasing power at this point, it should be possible to freely change your age, gender, add extra arms, do just about anything that is feasible for a creature still made out of meat.
b. Antimatter-fueled starships, assuming no hidden showstopper rules in the laws of physics regarding antimatter
c. Self replicating robotics, to the point that tearing down planets for raw materials is a straightforward, routine operation
d. People talk about becoming a “type N” civilization but this misses the point, the limiting factor wouldn’t be energy, it’s the matter available around the sun. Most likely most of the energy emitted by the sun would still be freely shining into space.
But yes, I cannot predict who would be in charge, if anyone, in such a world. Can’t even predict if there would even be intelligent beings like we call intelligence. Robotic agents that don’t have emotions, can’t experience art or music or sex or appreciate colors or sounds, but just ruthlessly copy themselves as quickly as possible, might be all that is left.