Several POD's
Massive population die-offs offer an apparently clean slate with no status quo to stop innovation, you'd think right?
Well, the big problem you've got when population drops by a massive chunk is who survived? A large # of little kids, old farts past reproducing, or a mixture of age ranges that just happen to be naturally immune?
In Case 1,
Lord of the Flies or
Earth Abides.
In this case, the kids don't get proper education, they focus so much on survival that expecting them to be rabid tinkerers and scientists who'd focus on tech advancement at all costs is nearly ASB to establish and maintain the base of techniques to keep civilization going IMO- agriculture, power generation, medicine, civil, chemical and mechanical engineering. Once you've got those, then progress can go crazy from there. If you don't, things decline for quite a while until humanity regains the nature-savvy to survive without the arsenal of tech that makes things easy it took us thousands of years and millions of casualties to amass.
Without agriculture creating enough food surplus so a proportion of folks can specialize and tinker, you're stuck with hunter-gathering which is very labor-intensive. Sure, there's plenty of canned food and Twinkies laying around, but I'd imagine those stocks would get heavily picked over pretty soon.
Population hits hunter-gatherer equilibrium @ 2-5% present population ~200 M.
In Case 2,
Bioshock, Logan's Run, or 13 Monkeys AKA Skinner Box society
You'd have a hell of a genetic bottleneck because barely 2% of the population can breed, for that first generation. The good news is, depending on what mix of folks survives, there's hopefully plenty of skilled, experienced folks to teach the first couple of generations of kids something useful and keeping much more of a veneer of sophistication going.
You'd see a huge reliance on robotics to do the dirty jobs (sparing the old farts from wearing themselves out growing food, mining necessary materials, keeping roads from falling apart) cloning and genetic engineering techniques to help replenish human population a lot more quickly.
I call it Rise of the Cylons after a few generations. Either the people get tired of being lab rats for the Elders to keep tweaking for giggles or a multi-generational agenda or the robots become self-aware and revolt.
Case 3
I look at this as
The Stand, The Road only without the obvious Good vs Evil.
Lots of variables in again, what sorts of folks survive, what skills they have and can develop with a little guidance, what opportunities they have to practice and improve their repertoire of skills, whether you have clear plans to rebuild civilization factions are committed to enacting or a bunch of scavengers looting and kludging together ever-less-effective tools as the spiffy gear wears out.
All this is assuming a massive GLOBAL die-off.
Some have mentioned local die-offs that societies bounced back from. The Black Death killed off 30-40% and was a social laxative to Western Europe. It didn't hurt that the Mongols and Muslim neighbors spurred the Europeans to mobilize to learn more about the outside world and get up to speed and in the process, taking Church doctrine and approval less seriously in a burst of brute pragmatism.
Most people put the Renaissance at 1500, where the Black Death occurred mostly in 1348-1350 with heinous, but localized outbreaks thereafter after herd immunity and responses to epidemics became more robust.
Any more than that and the abilities for a society to flourish afterwards are very slim indeed based on the archeological/historical evidence.