Let $X$ be smooth variety defined over $\mathbb{Q}$. If we want to check that $X$ is locally soluble at a prime $p$, then it suffices to find a non-singular $\mathbb{F}_p$ point, which can be lifted to a $\mathbb{Q}_p$ point by Hensel's lemma.

However, it might happen that $X$ does not have any non-singular $\mathbb{F}_p$ points. For example $X$ could be given by a quadratic form and we are interested in the prime $p=2$, in which case the reduction mod $2$ of $X$ is a non-reduced scheme and hence every point is singular (at least if I understand the situation correctly).

What general methods are there, if any, to check local solubility in this kind of situation?

A nice toy example is the equation $x^2 + y^2 + z^2=0$. This is locally soluble for all primes $p>=3$ (by Chevalleyâ€“Warning) and clearly not soluble for $p=\infty$, hence it is not soluble at $p=2$ (by the Hilbert symbol formulation of quadratic reciprocity). Is there a simple way to see this using general methods?