The guidelines do not clearly specify how much work one is assumed to do in writing a MathSciNet review. What is the consensus? Is one expected to read the whole paper and check the the proofs in detail?
2 Answers
You are not expected to check the proofs. You are expected to write an informative text which will reflect the contents of the paper. So that the reader can get an idea what did the author prove (or claims to prove), and decide whether s/he wants to read the paper. Of course, if you find a mistake you mention this. Or if you know that the result is not new. But you are not supposed to check the proof: it is the job of the referee.

8$\begingroup$ Regarding the last sentence that it's the job of the referee(s), here are varying opinions found on MO: mathoverflow.net/questions/40729/… $\endgroup$ Jan 5, 2015 at 17:11

3$\begingroup$ @YuichiroFujiwara, indeed, in my experience, some reputable journals explicitly say that ensuring "correctness" is not the responsibility of the referee, but of the author. Correctness of proofs is not the same as correctness of conclusions, which potentially adds to the confusion. $\endgroup$ Jan 5, 2015 at 17:19

7$\begingroup$ So let us say unambiguously that checking the proof is the job of the author, with an optional assist from the referee. :) $\endgroup$– LSpiceJan 5, 2015 at 18:01

3$\begingroup$ Whose responsibility is it to engage with postpublication identification of systemic errors? $\endgroup$ Jan 5, 2015 at 18:19

12$\begingroup$ @Adam Epstein: It is nobody's responsibility:) When you use some published result, it is only YOUR responsibility to check that it is correct. $\endgroup$ Jan 5, 2015 at 18:56
I echo the answer that you are not expected to check the proofs as a MathSciNet reviewer. That said, I personally choose to do so, because I believe that this leads to better reviews. When I read a review I want it to tell me what to expect from the paper, and if there is a gaping hole I want to know that fact before I spend hours stuck on it. Of course, this is assuming that the paper is of somewhat moderate to small length, which has often been the case in the papers I've reviewed. I also feel like putting my name on a review gives it my stamp of approval, and for me that stamp means something more than just looking over a paper quickly.
It might surprise you that I've found many errors in papers I've been asked to review. Some unfixable and central to the paper, some small and unfixable, lots of minor typos (which I don't mention in my reviews unless there are just gobs of them), and in one case an error that was quite subtle and it took a good week for me to find a fix. (I added the fix to my review, but only at the recommendation of the author of the paper.) Of course, you better be sure there is an error (perhaps by communicating with the author if appropriate) before mentioning it in a review. In all these cases, I hope that the people who read my reviews find them helpful.