When $S=\mathbb{P}^2$ and $\mathcal{L}= H= \mathcal{O}_{\mathbb{P}^2}(1)$, this question is discussed in Sernesi's book Deformation of Algebraic schemes, Chapter 4. In particular Corollary 4.7.19 page 266 states what follows:

(1) For every $d \geq 2$ and $0 \leq \delta \leq {d \choose 2}$ the Severi varieti $\mathcal{V}_{dH, \delta}$ is nonempty.

 

(2) For every $d \geq 2$ and $0 \leq \delta \leq {d-1 \choose 2}$ the Severi varieti $\mathcal{V}_{dH, \delta}$ contains irreducible curves.

These results have been partially extended on arbitrary projective surfaces. See again Sernesi's book, page 268 and the references given therein.

When $S=\mathbb{P}^2$ and $\mathcal{L}= H= \mathcal{O}_{\mathbb{P}^2}(1)$, this question is discussed in Sernesi's book Deformation of Algebraic schemes, Chapter 4. In particular Corollary 4.7.19 page 266 states what follows:

(1) For every $d \geq 2$ and $0 \leq \delta \leq {d \choose 2}$ the Severi varieti $\mathcal{V}_{dH, \delta}$ is nonempty.

(2) For every $d \geq 2$ and $0 \leq \delta \leq {d-1 \choose 2}$ the Severi varieti $\mathcal{V}_{dH, \delta}$ contains irreducible curves.

These results have been partially extended on arbitrary projective surfaces. See again Sernesi's book, page 268 and the references given therein.