Take $X = X_1\cup\dots\cup X_k\subset\mathbb{P}^n$ be the union of $k$ distinct smooth hypersurfaces with $\deg(X_i)\geq 2$ for $i = 1,\dots,k$, and $d = 1$. For any point $p\in X_i$ then intersection $T_pX\cap X$ is singular at $p$. Therefore, the dual hypersurface $X_i^{*}$ is a component of $S_1$.
Now, if $X_{i}^{*} = X_{j}^{*}$ then $X_i = X_i^{**} = X_j^{**} = X_j$. Then $X_i^{*} \neq X_j^{*}$ if $i\neq j$, and $S_1$ has at least $k$ irreducible components.
The irreducibility of $S_1$ holds if $X$ is smooth since the dual variety of a smooth variety is irreducible.