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In this paper, addressing Heinonen's ICM 02 talk, we study the geometry of the branch set $\\mathcal{B}_f$ of a quasiregular mapping between metric $n$-manifolds. In particular, we show that $\\mathcal{B}_f\\cap \\{x\\in X:H_f(x)<\\infty\\}$ is countably porous, as is its image $f\\big(\\mathcal{B}_f\\cap \\{x\\in X:H_f(x)<\\infty\\}\\big)$. 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