I am running several linear mixed models for an study about birds with the variable nest as a random variable. The thing is that in some of these models I get what is called \'s
Are you actually interested in whether each of the fixed effects in your model has an effect? For example, age or sex may explain some of the variation, but perhaps you could include it as a random effect rather than a fixed effect. Changing it to a random effect (if that is rational) might address the over dispersion issue.
My interpretation of the singularity issue, which certainly could be incorrect, is that each of the combinations of your model only has one observation/measurement. Therefore, you may not have enough observations to include all of those variables as fixed effects.