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Many popular texts that discuss evolution and natural selection often mention that many (or most) mutations are bad (not adaptive).

Have there been any studies on what the rough percentages are? (E.g. Is it 90%? 99%? 99.9%? If this number varies by species, an answer could just focus on one particular species.)

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    $\begingroup$ It would depend dramatically on where the starting point is on the fitness landscape. If it's on a peak, all mutations that aren't neutral are bad. If it's at a low point on the fitness landscape, more mutations will be adaptive than, e.g., mutations when the starting point is relatively high up the slope of a fithess "hill". $\endgroup$
    – S. McGrew
    Jun 12 '20 at 16:47
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As suggested by the McGrew comment, this is dramatically dependent on the shape of the fitness landscape. It will vary quite a bit from situation to situation.

More precisely, you are interested in the distribution of fitness effects (DFE) of mutations, and the upper tail specifically.

Wikipedia has a section on this. In there, they note one study of a virus that suggests that ~4% of mutations are beneficial. That may be an overestimate, and many of those are probably only very slightly beneficial. Probably the number is lower in non-viral organisms.

For a review on the DFE of new mutations, see here. They write in their section "Advantageous mutations":

As expected, relatively few of the mutations that are not effectively neutral are advantageous.In three mutagenesis experiments, the proportion of advantageous mutations was 4% in the RNA virus vesicular stomatitis virus (VSV)15 (FIG. 1), 0% in Escherichia coli(14), 0–15% in the bacteriophage φX174 (REF. 40), 0% in φ6 (REF. 13) and 6% in Saccharomyces cerevisiae (16).

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It is necessary to distinguish mutations from substitutions: assuming for simplicity that a mutation can happen at any place in the genetic sequence, most mutations are bound to result in non-functional genomes. Substitutions, on the other hand, are mutations that resulted in viable organisms - they can still have negative fitness effect, but not outright deleterious.

Note also that non-adaptive mutations are not necessarily bad - the neutral theory of evolution tells us that, due to randomness effects, substitutions may fix in the population without having a direct fitness advantage.

In viruses the quantity of substitutions can be very high - tens of percents, to the point that one has to introduce ad-hoc thresholds to distinguish viral strains, e.g., viruses that differ by 30% of their genetic content. How much of this is adaptive depends on the shape of the fitness landscape, as noted by @MaximilianPress. E.g., for HIV it has been found that up to a third of the sequence changes tend to revert to the ancestral HIV sequence, i.e., constitute host-specific adaptation.

Clarification
In view of the discussion that followed, I would like to add some precisions to my answer:

  • My distinction between mutations and substitutions is essentially distinction between the actual and the observed mutations. This is different from the more common usage, where substitution means point mutation, with mutation being a more general term.
  • Since I mentioned the neutral theory, it is necessary to remark that the entirety of its claims is not generally accepted and/or supported by data. Moreover, its validity depends on the organisms in question. My use of the neutral theory is therefore limited to the fact that being adaptive/non-adaptive is not the only factor that determines that a mutation fixes in the population.
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    $\begingroup$ According to whose definition?! It’s neither a very common nor a particularly useful definition. It’s much more common that “mutation” refers to any change of genetic sequence, and “substitution” refers to the exchange of a single nucleotide (it’s a type of mutation, same as e.g. indels or rearrangements). Neither terms say anything about the change’s effect on fitness. $\endgroup$ Dec 17 '20 at 12:30
  • $\begingroup$ @KonradRudolph Indeed, there are no unique agreed upon definitions for either term. Which is why I explicitly explained what I mean. $\endgroup$ Dec 17 '20 at 13:06
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    $\begingroup$ Hmm I disagree with the substance of that. There are widely agreed definitions, and they are different from the ones you are using, which needlessly causes confusion. And the distinction you introduce doesn’t help answer OP’s question, it’s completely irrelevant here. — Another problem with this answer is that the neutral theory of evolution is extremely controversial and, to put it bluntly, probably untrue (or, if it happens to be true in some extremely narrow sense, it makes no useful predictions). $\endgroup$ Dec 17 '20 at 13:19
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    $\begingroup$ Oh, I agree that that distinction is indeed important for answering OP’s question, I didn’t get that this is what you’re referring to from your answer. — And regarding my statement of “controversial”, I was being diplomatic. There is effectively zero evidence in favour of the hypothesis, and modern statistical evidence is pretty one-sidedly against it. The way the hypothesis is nowadays “defended” is by diluting it so far that it makes no predictions. $\endgroup$ Dec 17 '20 at 13:31
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    $\begingroup$ @KonradRudolph I added clarifications to accomodate your comments. $\endgroup$ Dec 17 '20 at 13:57

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