The degeneracy locus can be reducible, and even non-reduced. For instance, take $X= \mathbb{P}^2$ and consider a morphism

$\mathcal{O}(-2)^2 \stackrel{f} \to \mathcal{O}^2$.

$f$ is given by a $2 \times 2$ matrix of forms of degree $2$, so its degeneracy locus is a quartic curve. For a general choice of $f$ this curve will be smooth, but for special choice of the matrix it can become singular or even a double conic.

The best results I am aware of can be found in Ottaviani's book "Varieta' proiettive di codimensione piccola" [projective varieties of small codimension, unfortunately I do not think an english translation is available].

Set

$D_k(f):=\{x \in X \; | \; \textrm{rank}(f_x) \leq k \}$

Then we have the following

Theorem (of Bertini's type)

Set $\textrm{rank}(E)=m$, $\textrm{rank}(F)=n$. Assume that $E^{*} \otimes F$ is globally generated. Then for the generic morphism $f \colon E \to F$, the locus $D_k(f)$ is either empty or it has the expected codimension $(m-k)(n-k)$, and the singular locus of $D_k(f)$ is contained in $D_{k-1}(f)$.

In particular, if

$\dim X < (m-k+1)(n-k+1)$

then $D_k(f)$ is smooth for a general choice of $f$.

The degeneracy locus can be reducible, and even non-reduced. For instance, take $X= \mathbb{P}^2$ and consider a morphism

$\mathcal{O}(-1)^2 \stackrel{f} \to \mathcal{O}^2$.

$f$ is given by a $2 \times 2$ matrix of linear forms, so its degeneracy locus is a conic. For a general choice of $f$ this conic will be smooth, but for special choice of the matrix it can become singular or even a double line.

The best result I am aware of can be found in Ottaviani's book "Varieta' proiettive di codimensione piccola" [projective varieties of small codimension, unfortunately I do not think an english translation is available].

Set

$D_k(f):=\{x \in X \; | \; \textrm{rank}(f_x) \leq k \}$

Then we have the following

Theorem (of Bertini's type)

Set $\textrm{rank}(E)=m$, $\textrm{rank}(F)=n$. Assume that $E^{*} \otimes F$ is globally generated. Then for the generic morphism $f \colon E \to F$, the locus $D_k(f)$ is either empty or it has the expected codimension $(m-k)(n-k)$, and the singular locus of $D_k(f)$ is contained in $D_{k-1}(f)$.

In particular, if

$\dim X < (m-k+1)(n-k+1)$

then $D_k(f)$ is smooth for a general choice of $f$.