Meaning of BWA-MEM MAPQ scores Does anyone know what the MAPQ values produced by BWA-MEM mean? 

I'm looking for something similar to what Keith Bradnam discovered for Tophat v 1.4.1, where he realized that:


  0 = maps to 5 or more locations 
  
  1 = maps to 3-4 locations 
  
  3 = maps to
  2 locations 
  
  255 = unique mapping


I'm familiar with the notion that MAPQ should be theoretically be related to the probability of an "incorrect" alignment (10^(-MAPQ/10)), although this is vaguely-specified enough that aligners in practice tend to actually just use something like the above. 

Note that this is different than the BWA MAPQ scoring interpretation, because BWA-MEM gives MAPQ scores in the range $[0,60]$, rather than $[0,37]$ as has been established for BWA.
First of all, if you want to understand mapping quality (mapQ), ignore RNA-seq mappers. They often produce misleading mapQ because mapQ is not important to RNA-seq anyway.

Strictly speaking, you have two questions, one in the title: the meaning of mapQ; and the other in a comment: how mapQ is computed. On the meaning, mapQ is nearly the same as baseQ – the phred scaled probability of the alignment/base being wrong. It often amuses me that we question mapQ but take baseQ for granted. BaseQ is also scaled and discretized differently; even fewer people know how Illumina/pacbio/nanopore/historical sequencers estimates baseQ.

On the second question, Section 2 of the MAQ supplementary explains the theoretical aspects of mapQ, which is still correct today. Briefly, mapping quality consists of three components: 1) the probability of contamination; 2) the effect of mapping heuristics and 3) the error due to the repetitiveness of the reference. Only 3) can be modeled theoretically.

In case of bwa-mem, if we assume the matching score is 1, type-3 error is estimated with:

$$10/\log10\cdot[\log4\cdot(S_1-S_2)-\log n_{\rm sub}]$$

where $S_1$ is the best alignment score, $S_2$ is the second best and $n_{\rm sub}$ is the number of suboptimal alignments. Factor $\log 4$ comes from the scoring matrix. Factor $10/\log10$ is the Phred scale. This equation assumes gap-free alignment and is very close to Section 2.5.2 in the MAQ supplementary. It is ok-ish for short reads, but often overestimates for long reads. I am not aware of a practical approach in general cases. In addition to this method, you can estimate mapQ by read simulation: just try to find a function such that it fits empirical mapQ. Some have tried machine learning, too.