I'd bet everything I own against a farthing that the singularity will never, ever happen no matter how long humanity exists. If, that is, there were any way for me to live long enough to collect.
As far as I am concerned, The Singularity belongs in the same category as perpetual motion machines - it simply can't happen.
"The Singularity" is misnamed. It should be "The Event Horizon". And we
might conceivably get to an Event Horizon. All it takes is for the momentum to build, devopments begetting further developments, and then suddenly it seems we're not the controlling part of the relationship. (I'm not predicting that this happens, I'm just saying that it isn't inconceivable.)
For code-breaking, above, which is merely one possible sidetrack that I'm not myself seriously suggesting will be the main impact, the problem is not that they have actual magical woo-woo that gets past logical limitations/speedbumps, but that:
a) The AI that learns to solve problems might apply itself to solving finding (e.g.) a novel method of obtaining the unique prime factors behind any given encryption layer, and may thus find the philosopher's stone it needs.
b) In the above process and/or the brute-forcing, an AI probably need not sleep, rest, take a break, need to double-check what avenues it has gone down (is currently going down, has tried and hit a wall with before) whenever it resumes again, etc. And if it's taking too much time, individually, it doesn't need to try to find help, hire employees/subordinates/assistants and vet the candidates for competency and trustworthiness to usefully outsource some of the work to. It might (depending upon what freedom it obtains to do such things) just be able to spawn a duplicate version ...or a specialist sub-process... to parallelise its work. And if it can do that once then it can do it again, again, again, making hundreds, thousands of times. Possibly even subverting millions of external machines in ways already doable for Botnets. Or cleverly designing something like a Folding@Home or Galaxy Zoo that actually socially engineers 'mere' humans into assisting in the work. Perhaps even fully 'gamifying' something... That new Tetris/Brick-breaker/Merge game that everyone's playing, is it perhaps getting players to contribute 'wetware' power-cycles to somehow implement a distributed sieve of Eratosthenes?
Even quantum-computing isn't the answer to completing cryptographic solutions. It still needs to be used correctly. And smart minds can develop better hardware, better ways of isolating quantum dots. Having created the hardware developing the technique, how do you shove things into it, how do you get it to mix it all up, what is supposed to be extracted from it in the end? People think they know how to do it (that's better than merely 'quantumly brute-forcing' the problem), and I suspect that an AI could apply its own "it will not stop, it will not sleep" ability to the issue of optimising it a, whether asked to do so by humans (defining parameters and letting a 'machine' refine the best answers) or deciding entirely by itself that this is a necessary step on its own particular Paperclip quest.
Of course, this does rely upon an actually sufficiently intelligent (and motivated, and resourced) AI, and I'm not sure we currently have anything like that - and likely still not in a position do be used in any such situation. But if you ever happen to set up just the right dominoes in just the right way then it takes just a single nudge to set the while lot in motion before you necessarily know what you've done.