I updated the notebook with explanations for this discussion with a new section where it is given and explained NIntegrate methods tracing code.

Here is an example output of the tracing (see the notebook for details):

In many cases I find it helpful to look at the sampling points used by NIntegrate. Below are some examples:

Needs["Integration`NIntegrateUtilities`"]

NIntegrateProfile[NIntegrate[1/Sqrt[x], {x, 0, 1}]]

NIntegrateSamplingPoints[NIntegrate[1/Sqrt[x], {x, 0, 1}]]

NIntegrateSamplingPoints[
 NIntegrate[1/Sqrt[x], {x, 0, 1}, 
  Method -> {"GlobalAdaptive", "SingularityHandler" -> "IMT"}]]

NIntegrateSamplingPoints[
 NIntegrate[1/Sqrt[x], {x, 0, 1}, 
  Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0, 
    "SingularityHandler" -> "IMT"}]]

NIntegrateSamplingPoints[
 NIntegrate[1/Sqrt[x], {x, 0, 1}, 
  Method -> {"GlobalAdaptive", "SingularityDepth" -> 20}]]

NIntegrateSamplingPoints[
 NIntegrate[1/Sqrt[Abs[x]], {x, -2, 1}, 
  Method -> {"GlobalAdaptive", "SingularityD> 2}]]

NIntegrateSamplingPoints[
  NIntegrate[1/Sqrt[Sin[x]], {x, 0, \[Pi]}, 
    Method -> {"GlobalAdaptive", "SingularityDepth" -> 2}]]

This is a good question.

These are related posts:

http://mathematica.stackexchange.com/questions/26401/determining-which-rule-nintegrate-selects-automatically http://mathematica.stackexchange.com/questions/41504/is-there-a-way-to-see-the-result-of-nintegrates-symbolic-preprocessing