Random Weierstrass equations over Z_p have Haar-measure densities for admitting a Q_p-rational 3-torsion point or 3-isogeny given by exact rational functions depending on p modulo 3.
Local and global densities for Weierstrass models of elliptic curves
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The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny
Random Weierstrass equations over Z_p have Haar-measure densities for admitting a Q_p-rational 3-torsion point or 3-isogeny given by exact rational functions depending on p modulo 3.