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We study some properties of the polar curve $P_l^{ℱ}$ associated to a singular holomorphic foliation $ℱ$ on the complex projective plane ${\mathbb{P}}^2$. We prove that, for a generic center $l\in {\mathbb{P}}^2$, the curve $P_l^{ℱ}$ is irreducible and its singular points are exactly the singular points of $ℱ$ with vanishing linear part. We also obtain upper bounds for the algebraic multiplicities of the singularities of $ℱ$ and for its number of radial singularities.