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We define the singular Hecke algebra $\mathscr{H}\left(S{B}_n\right)$ as the quotient of the singular braid monoid algebra $ℂ\left(q\right)\left[S{B}_n\right]$ by the Hecke relations ${\sigma }_k^2=(q-1){\sigma }_k+q$, $1\le k\le n-1$. We define the notion of Markov trace in this context, fixing the number $d$ of singular points, and we prove that a Markov trace determines an invariant on the links with $d$ singular points which satisfies some skein relation. Let ${\mathrm{TR}}_d$ denote the set of Markov traces with $d$ singular points. This is a $ℂ(q,z)$-vector space. Our main result is that ${\mathrm{TR}}_d$ is of dimension $d+1$. This result is completed...