black holes - Curvature Invariants in General Relativity and Singularities

Suppose that I want to check if a given metric is singular or not. I'm interested in curvature singularities, not coordinate singularities, so I can look to scalars made with Ricci, Riemann and Weyl Tensor.

If I found that one of this scalar is divergent somewhere, then I'm done. My problem is the opposite, suppose that I don't find singularities after checking some invariants. How can I be sure that the space is non singular? Rephrased: Is there a COMPLETE basis of scalar curvature invariants in general relativity? Let's say in $D=4$ for concreteness. The vacuum case in particular.

I heard somewhere that in vacuum and in $D=4$ is enough to restrict to: $R$, $R_{\mu \nu \rho \sigma} R^{\mu \nu \rho \sigma} $, $R_{\mu \nu } R^{\mu \nu } $, ${{}^\star R}_{\mu \nu \rho \sigma} R^{\mu \nu \rho \sigma} $, $ {{}^\star R}^{\star}_{\mu \nu \rho \sigma} R^{\mu \nu \rho \sigma} $. Is this true?

References are welcome.

EDIT: To be more precise, referring only to curvature singularities (I know that there are other way to characterize a singularity like explicitly working with geodesics) is there a minimum number of invariants to check, in order to conclude that the metric is free of curvature divergencies?