Let $S$ be a smooth, projective surface over $\mathbb{C}$ and $L\in\mathrm{Pic}(S)$ be globally generated. Then, a general curve $C\in\vert L\vert$ is smooth. Now, let $I_\xi$ be the ideal of a (possibly nonreduced) $0$dimensional subscheme $\xi\subset S$ and assume that $L\otimes I_\xi$ is still globally generated. Which hypotheses on $\xi$ would ensure the existence of a smooth curve in $\vert L\otimes I_\xi\vert=\mathbb{P}(H^0(S,L\otimes I_\xi)$?
$\xi$ has to be "curvilinear", i.e., contained in some smooth curve of $S$, or equivalently, has to be of multiplicity one. It is clear that this is necessary. Let me now show that it is sufficient. The simplest case is when $\xi$ is reduced. Then you consider the blow up $\pi_\xi:\tilde S\rightarrow S$ at all points in $\xi$, and notice that $L\otimes I_\xi\cong \pi^*LE_\xi$. General curves in $\pi^*LE_\xi$ are smooth and intersect $E$ transversely, because $\pi^*LE_\xi$ is globally generated, so you are done. Now assume the curvilinear scheme $\xi$ is nonreduced but irreducible of length $k$ supported at some point $p=p_1$, and consider the blow up $\pi_1: S_1\rightarrow S$ at $p$, with exceptional divisor $E_1$. Then $\pi_1^*I_\xi=I_{E_1}\cdot I_{\xi_1}\cong I_{\xi_1}\otimes \mathcal O_{S_1}(E_1)$, where $\xi_1$ is a 0dimensional curvilinear subscheme of $S_1$, irreducible of length $k1$, supported at some point $p_2\in E_1$. More important, $I_\xi=\pi_{1*}(I_{\xi_1}\otimes \mathcal O_{S_1}(E_1))$, so $I_{\xi_1}\otimes \mathcal O_{S_1}(E_1)$ is globally generated. Blow up $p_2$ to define $p_3$, etc, till you blow up $p_k$; denote $\pi:\tilde S \rightarrow S$ the composition of the $k$ blowups, and let $E_i$ be the pullback on $\tilde S$ of the exceptional divisor above $p_i$. Now $I_\xi=\pi_*(\mathcal O_{\tilde S}(E_1\dotsE_k))$ with $\mathcal O_{\tilde S}(\pi^*LE_1\dotsE_k)$ base point free. A general member of $\pi^*LE_1\dotsE_k$ is smooth, intersects $E_k$ transversely at a point different from the singular point of $E_1$, so its image on $S$ is nonsingular. The above analysis can be carried over for each point on the support of $\xi$. A local version of this, including the description of what happens for noncurvilinear subschemes (the singularity type of a general member of $L\otimes I_\xi$ is determined by the "resolution" of $\xi$), can be found (as "Bertini theorem") on CasasAlvero's book on singularities of plane curves. 

