What exactly is meant by a singular fit of a mixed model, and why does it result in perfect correlations among random effects? I understand a singular fit to be cases where a random effect has a variance of 0. Does this essentially mean that the model could not find a variance parameter for the random effect that did better than 0? Why is this called a "singular" fit? Does it refer to there only being a "single" value for each subject's intercept, for example?
These singular fits often come along with perfect correlations among random effects. What exactly causes these perfect correlations? That is, what exactly happens in the estimation process that leads to random effects that are perfectly redundant?
Lastly, is this a failure to converge? Or is that a separate issue?
 A: *

*For scalar random effects (i.e. random effects with only one term, where the estimated parameter is a single variance value) a singular fit indeed means that the variance is zero.

*For bivariate random effects such as a random slopes term, we are estimating a 2 $\times$ 2 covariance matrix; a singular fit may manifest itself as an estimate of zero for either variance (in which case the correlation is undefined), or a correlation of +/- 1.

*For more random effects with dimension >2, a singular fit need not show up as either variances equal to zero or correlations equal to +/-1.


A definition of a singular matrix from MathWorld:

A square matrix that does not have a matrix inverse. A matrix is singular iff [if and only if] its determinant is 0.

From ?lme4::isSingular:

Complex mixed-effect models (i.e., those with a large number of
       variance-covariance parameters) frequently result in singular
       fits, i.e. estimated variance-covariance matrices with less than
       full rank. 

["Less than full rank" is synonymous with the definition above]

Less technically, this means that some "dimensions" of
       the variance-covariance matrix have been estimated as exactly
       zero. For scalar random effects such as intercept-only models, or
       2-dimensional random effects such as intercept+slope models,
       singularity is relatively easy to detect because it leads to
       random-effect variance estimates of (nearly) zero, or estimates of
       correlations that are (almost) exactly -1 or 1. However, for more
       complex models (variance-covariance matrices of dimension >=3)
       singularity can be hard to detect; models can often be singular
       without any of their individual variances being close to zero or
       correlations being close to +/-1.

Failure to converge is a separate issue; it means the optimizer thinks it might not have found a best fit, not that the best fit didn't correspond to a singular covariance matrix.
