Sauron decided to make nineteen rings of power for elves, dwarves and men - and the One Ring to control them. So, did his plan work?
Overall not. The nine rings for men were the best success; since they produced the Ringwraiths, who were powerful and completely loyal to their master. But the dwarves were resistant to being enslaved - so Sauron got those back which had not been destroyed by dragon fire. And the elven rings were never tainted by the touch of Sauron, and were only used when he did not have the One Ring.
Even when Sauron had possession of the One Ring - and maximally enhanced by it; nonetheless, his armies were defeated by the Last Alliance of elves and Men - and Sauron himself was slain (in his bodily form) by Gil-galad and Elendil.
And then, after Sauron had lost possession of the One Ring; he was destroyed spiritually by its destruction.
If we presume that the purpose of the rings of power was to gain power for Sauron - because power was Sauron's primary goal; then they did not work; and did him more harm than good in the end.
But if the rings were an error from the point of view of Sauron's will for power; they were a success from the point of view of Morgoth's 'Satanic' agenda; which was to corrupt the world and creation, in spite of Eru.
The corruption of the Ringwraiths has always been mentioned. But although not enslaved by their rings, the dwarves were corrupted. The rings enhanced their core sin - which was greed for material things, for wealth. The dwarf rings were desired by (some) dwarves because they were supposed to amplify wealth, that was the flaw of greed which the rings then amplified.
And even the elf rings amplified the elves major sin - which was the desire to slow down time and change: to make the mortal lands of middle earth more like the undying lands. This increased the elvish tendency to inertia, regret; towards passive self-absorption.
Plus the One Ring, after Sauron had lost it. This caused considerable and widespread corruption. Most obviously by turning Saruman (the greatest wizard) to the side of evil. Also the ill effects on Isildur, Denethor and Boromir (among the greatest of Men). Then there was the chronic temptation of Galadriel - the greatest High Elf of the age (eventually overcome).
At the smaller scale there were historically significant effects on the proto-hobbits Deagol and Smeagol - and the pain and corruption done to Frodo.
Indeed, only Bilbo and Sam seem to have had net-benefit from the One Ring.
So the rings of power failed to achieve dominating power for their maker Sauron; but achieved an increase in the amount of evil in the world which benefitted Sauron's master - Morgoth. They made a considerable contribution to the agenda of evil.
Acknowledgment: The above ideas were developed in conversation with my son, Billy.