After all, if a paper has taken years to write and has gone through rigorous peer review, then surely the statistics are going to be rock solid?
My experience of reading papers that attempt to apply statistics across a wide variety of areas (political science, economics, psychology, medicine, biology, finance, actuarial science, accounting, optics, astronomy, and many, many others) is that the quality of the statistical analysis may be anywhere on the spectrum from excellent and well done to egregious nonsense. I have seen good analysis in every one of the areas I have mentioned, and pretty poorly done analysis in almost all of them.
Some journals are generally pretty good, and some can be more like playing darts with a blindfold - you might get most of them not too terribly far off the target, but there's going to be a few in the wall, the floor and the ceiling. And maybe the cat.
I don't plan on naming any culprits, but I will say I have seen academic careers built on faulty use of statistics (i.e. where the same mistakes and misunderstandings were repeated in paper after paper, over more than a decade).
So my advice is let the reader beware; don't trust that the editors and peer reviewers know what they're doing. Over time you may get a good sense of which authors can generally be relied on to not do anything too shocking, and which ones should be treated especially warily. You may get a sense that some journals typically have very high standard for their stats.
But even a typically good author can make a mistake, or referees and editors can fail to pick up errors they might normally find; a typically good journal can publish a howler.
[Sometimes, you'll even see really bad papers win prizes or awards... which doesn't say much for the quality of the people judging the prize, either.]
I wouldn't like to guess what the fraction of "bad" stats I might have seen (in various guises, and at every stage from defining the question, design of the study, data collection, data management, ... right through to analysis and conclusions), but it's not nearly small enough for me to feel comfortable.
I could point to examples, but I don't think this is the right forum to do that. (It would be nice if there was a good forum for that, actually, but then again, it would likely become highly "politicized" quite quickly, and soon fail to serve its purpose.)
I've spent some time trawling through PLOS ONE ... and again, not going to point at specific papers. Some things I noticed: it looks like a large proportion of papers have stats in them, probably more than half having hypothesis tests. The main dangers seem to be lots of tests, either with high $\alpha$ like 0.05 on each (which is not automatically a problem as long as we understand that quite a few really tiny effects might show up as significant by chance), or an incredibly low individual significance level, which will tend to give low power. I also saw a number of cases where about half a dozen different tests were apparently applied to resolving exactly the same question. This strikes me as a generally bad idea. Overall the standard was pretty good across a few dozen papers, but in the past I have seen an absolutely terrible paper there.
[Perhaps I could indulge in just one example, indirectly. This question asks about one doing something quite dubious. It's far from the worst thing I've seen.]
On the other hand, I also see (even more frequently) cases where people are forced to jump through all kinds of unnecessary hoops to get their analysis accepted; perfectly reasonable things to do are not accepted because there's a "right" way to do things according to a reviewer or an editor or a supervisor, or just in the unspoken culture of a particular area.
Best Answer
In mid 2000s, a group of medical statisticians put their heads together and issued STROBE statement (http://www.strobe-statement.org): STrengthening the Reporting of OBservational studies in Epidemiology. It was published in the same form in Lancet, PLoS Medicine, Journal of Clinical Epidemiology, and several others, which to me seems like the most amazing part of the whole exercise: putting heads together is not nearly as difficult as convincing a diverse group of editors to publish anything as is. There are various checklists based on the STROBE statement that help you define what a "well written" statistical part is.
In an unrelated area, the U.S. Institute of Education has been accumulating evidence on performance of the various educational programs in their What Works Clearinghouse. Their Procedures and Standards Handbook delineates what constitutes a solid study (by the education community standards; biostatisticians with clinical trials background find them falling quite short of what FDA requires). Spoiler alert: of the 10,000 study reports in the WWC data base, only 500 "meet WWC standards without reservations"... so when you hear somebody say about an educational product that it is "research-based", there's exactly 95% chance it's actually bogus, with research conducted by the publishers of that product without the control group.