This is a very soft question, and I'm not sure what I expect as an answer.

In SGA6, Expose XIII, Theoreme 5.1 it is proven that, if $X$ is a proper scheme over a field $k$, then $NS(X)$ is finitely generated. Here $NS(X) := \mathrm{Pic}_{X/k}(k)/\mathrm{Pic}_{X/k}^0(k)$.

However, on wikipedia's page for the "theorem of the base" this theorem is only stated for **smooth** projective varieties. In an earlier question on MO the same happens; see Modern Proof of the Theorem of the Base. A quick google search gives a lot of papers citing this result from SGA6, but most authors impose smoothness when citing SGA6. Do note that Section 5.3 in these notes math.stanford.edu/~conrad/249CS15Page/handouts/abvarnotes.pdf states that $NS(X)$ is finitely generated for proper geometrically integral, not necessarily smooth, schemes.

What is the reason that the theorem of the base from SGA6 is mostly cited for **smooth** proper schemes? Does the Neron-Severi group exhibit pathological behaviour if $X$ is singular?

Neron Modelsthe result is stated in the same generality as in SGA6 (in particular, no parasitic smoothness hypotheses). $\endgroup$