I'm wondering if there are general techniques to calculate the singular cohomology groups of a fiber space (specifically, a non-smooth elliptic fibration) using methods of algebraic topology. More concretely:

Suppose $X, Y$ are smooth, proper irreducible varieties over $\mathbb{C}$, and $X \xrightarrow{\phi} Y$ is a smooth map.

$\textbf{Question 1:}$ Do the singular cohomology groups of $X$ (let's say $\mathbb{Q}$-coefficients) correspond to the cohomology groups of a locally constant sheaf on $Y$?

$\textbf{Question 2:}$ Suppose we don't require that $\phi$ is smooth, but say instead that it just has reduced fibers. Is the same true? Can we even recover $\mathbb{Z}$-coefficients in some cases?

From what I understand, we can compute cohomology of the derived pushforward of the constant sheaf $\mathbb{Z}_X$, and this will give the singular cohomology groups of $X$, but now calculated on $Y$ (however, this is a tautological statement, and I don't really see how it helps with computations). If in our cases, the derived pushforward had the same cohomology as the sheaf $\phi_*\mathbb{Z}_X$ then we we would be done. References will be greatly appreciated!