- 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.
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
["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.