  Consider the integral polynomials $f_{a,b}(x)=x^4+ax+b$ and
$g_{c,d}(x)=x^4+cx^3+d$. Suppose $f_{a,b}(x)$ and $g_{c,d}(x)$ are irreducible,
$b\mid a$, and the integers $b$, $d$, $256d-27c^4$, and
$\dfrac{256b^3-27a^4}{\gcd(256b^3,27a^4)}$ are all square-free. Using the
Montes algorithm, we show that a root of $f_{a,b}(x)$ or $g_{c,d}(x)$ defines a
monogenic extension of $\mathbb{Q}$ and serves as a generator for a power
integral basis of the ring of integers. In fact, we show monogeneity for
slightly more general families. Further, we obtain lower bounds on the density
of polynomials generating monogenic $S_4$ fields within the families
$f_{b,b}(x)$ and $g_{1,d}(x)$.
