Does this mean the warning is a "false positive" and can safely be ignored ?
A singular fit is quite specifically defined, at least in
lme4, which I assume is what you are using.
The warning in
lme4 comes from a principal components analysis of the variance-covariance matrix of estimated random effects. If this is not of full rank, then the fit is singular. This is the definition of a singular fit. Correlations close to +/- 1 or variances close to zero, are very common symptoms when the random structure is simple, but when it is complex, these symptoms are not always present.
A model that has a singular fit is an over-fitted model so it is not a good idea to ignore the warning. For one thing, it will not be generalisable, but also, at least in my experience, such models can always be simplified, and result in much more parsimonious models that are easier to interpret. They are also often underpowered.
It is worth bearing in mind that correlations of +/- 1 are possible, but we should take a moment to think what that means.If we had such correlations in our observed data, then our fixed effects model matrix would be rank deficient and our model would be unidentifiable. Common sense would prevail and we would remove one of the variables from the model. The same applies to the random effects structure. Similarly, it is certainly possible that a variance for a random effect is zero. If so why would we wish to retain it ?