Well, the two big objections I'm used to hearing that are usually pretty legit where NS is concerned - as opposed to weird ass nonsense like 'violates laws of thermodynamics' - is that falsifiability thing and the tautology accusation. Those are reasonably fair in some contexts, particular if the context is someone engaging in sloppy definitions. "Survival of the fittest', for instance, is a tautology when one defines fitness as survivability. So one does have to be a bit careful with phrasing to avoid that and I'd guess most people - people not scientists - do actually have a flawed idea of NS that is a tautology, or maybe just a rhetorical tautology, not really a mind reader.

The falsifiability issue is a little rougher. Let me see if I can pull an example out of my ass. Take two coins, one weighted to come up heads 60% of the time, 60-40, and the other the normal 50-50. If the only way I have to test them is seeing them flipped I have a problem, and with NS most of our evidence is effectively constrained in a similar fashion, we can mostly only see the flips, so to speak. If I flip both coins 10 times, and get one 6-4 and the other 5-5 I can't declare the 6-4 coin to be the biased one, not with much confidence anyway. A normal coin will come up 6-4 in 10 flips very regularly, ditto 5-5 for the weighted coin. If I slip them 100 times and one comes up 59-41 and the other 51-49, I still can't be sure, but I can say which is which with greater confidence, still not too much though, because neither is really and abnormal result for either coin. There is literally no number of flips, except infinite, that can establish which coin is which so flipping coins to check for bias is not an experiment that includes falsifiability. No number of flips of the weighted coin can ever prove it is weighted. Classic Popper comments, "All men are mortal" is not falsifiable, since there are people who aren't currently dead, whereas "All men are immortal" is falsifiable, since they are currently people who are dead. Then of course some things are falsifiable but not practically falsifiable, "Some men do not pick their noses" is just not something you can realistic expect to check. Same as our coin, we check these things by observing them and seeing how often they occur then expressing the result with a certain confidence, "We have observed 1000 people each for 1000 hours via hidden video at purely random times and places. 967 of them picked their noses a total of 178,952 times, none of the pickers did it less than 15 times, we therefore conclude with X amount of confidence that the other 33 people do not pick their noses and any nose-picking by them is likely a trivial rarity, 3.3% of the population, plus or minus x%, do not pick their noses, with y% confidence". That's perfectly scientific you just have to make sure whatever conclusion you draw reflect that and say that. "This is the weighted coin, I'm 99.999....99% sure, as I flipped it some billion times" etc.

This can be made a lot worse though, two examples. First, regarding our tautology issue... if I deliberate breed several generations of animals for some horrendously bad trait, like a 10 pound poodle with a 5 foot tail, I can point at it and say 'why does it exist?' and you can answer something like that by pointing out that natural selection did not occur because I was screwing with it... or you can say 'wait, you, yourself, became the dominant control mechanism, a stupidly long and useless tail was a beneficial trait'. This is actually troublesome philosophically either way but you can argue, fairly reasonably, that any definition of NS that includes my freaky dog breed as a product is a trivial one, effectively useless, or leaves one stuck trying to cherry-pick a suitable definition of 'natural'. This is genuine concern though because or real knowledge of previous species is mostly limited to the fossil record, which is not really that large, and even using Occam's Razor ferociously we can't not just rule out something fairly absurd like a species that got kicks breeding animals for dumb traits but something more reasonable like a species developing a beneficial trait of hunting animals that were generally healthy as determined by being, say, fairly symmetric, thus killing off all the healthy members and leaving the barely useful mutants alive to procreate, but more or less eliminating predation from other sources simply by being around. That latter is no problem, really, except that it's really next to impossible to test things for falsifiability when you're in that region and you also have a problem of coming around to asking "What damn good is this theory, from a practical standpoint, if it boils down nothing firmer than 'stuff happens'" or again our tautology example, survival of the fittest where fit is defined, totally uselessly and circularly, as survivability.

The other one is quantum, which can be ignored almost always but fucks everything up all the time. By quantum, abiogenesis is not a problem, just an improbability, same as a spontaneous rain of snicker's bars. It would be wrong to say that with quantum, anything is possible, but there's precious little you can rule out. If I show that abiogenesis taking place by some non-quantum chicanery is X% and by quantum is Y%, then X >> Y would establish, to a certain confidence, that that is what happened... it would not be falsifiable any more than that coin example was. The problem of course here is that we have no idea what X is, I remember Hoyle calculated it to be some absurdly tiny number 10^-100 or other, flawed analysis though, but nobody's to my knowledge ever produced a solid figure. This too isn't too big a problem, we know a tornado can sweep through a junkyard and assemble an airplane by freak accident, no quantum involved even. If we find a plane in a junkyard while we can't rule out tornado assembly or quantum fluke, we can say other explanations - accidental mis-shipment of a working plane instead of the broken one next to it in the hangar, or a worker at the junkyard who assembled the plane out of scrap as a hobby - are vastly, vastly more likely scenarios than the tornado or QM fluke. That doesn't mean they can absolutely be ruled out though, nor does it mean mis-shipment or hobbyist are the correct answers. If all you've got for evidence, one way or another, is a photo of a plane in a junkyard, your not really in a position to verify any of those options and you also need to ask whether or not any answer you come up with is actually useful in addition to verifiable.

I hope that helped a bit, I'm assuming through out this that you're not really particular skeptical about evolution, I myself am skeptical about it merely from general contrariness and totally support the theory, but are more interested in the general philosophical aspects of this sort of thing, so I wrote it accordingly. Biology though is not my strong point and all the examples above are strictly off the top of my head. But the core point is that, especially where science is being used for some useful end, falsifiability is totally acceptable and hardly unscientific, so there's nothing wrong with evolution if bits of it aren't falsifiable. If a hundred flips of each coin is all you get before you have to go use that coin to decide who wins a million-dollar bet, the confidence value is high enough, being better than nothing.