Is every projective $\mathbf{Z}[x]$-module free? Is every finitely generated projective $\mathbf{Z}[x]$-module free?
 A: While it's certainly true (per Fernando's comment) that this is a special case of the Quillen-Suslin theorem, it was certainly known long before Quillen and Suslin came along.  
There's a paper of Murthy from the mid-1960s which shows that every projective $R[x]$-module is extended whenever $R$ is a regular ring of dimension at most 2.  ("Extended" here means "of the form $P[x]$ where $P$ is a projective $R$-module".  Since all projective ${\mathbb Z}$-modules are free, extended is equivalent to free in this case.)  
But there's an even earlier paper of Bass which covers the case where $R$ is regular of dimension 1, which is all you need.    The paper is called "Torsion Free and Projective Modules".
Edited to add:  And the case of a PID predates even Bass; I think it's due to Seshadri in the 1950s.
A: When $R$ is a PID, then every finitely generated projective $R[x]$-module is free. As Steven already said, this is due to Seshadri. Here is the reference:

Seshadri, C.S., Triviality of vector bundles over the affine space $K^2$, Proc. Nat. Acad. Sci. USA 44 (1958), 456-458.

