Summary: This article studies typical 1-Lipschitz images of \(n\)-rectifiable metric spaces \(E\) into \(\mathbb{R}^m\) for \(m\geq n\). For example, if \(E\subset\mathbb{R}^k\), we show that the Jacobian of such a typical 1-Lipschitz map equals 1 \(\mathcal{H}^n\)-almost everywhere and, if \(m>n\), preserves the Hausdorff measure of \(E\). In general, we provide sufficient conditions, in terms of the tangent norms of \(E\), for when a typical 1-Lipschitz map preserves the Hausdorff measure of \(E\), up to some constant multiple. Almost optimal results for strongly \(n\)-rectifiable metric spaces are obtained. On the other hand, for any norm \(\vert\,{\cdot}\,\vert\) on \(\mathbb{R}^m\), we show that, in the space of 1-Lipschitz functions from \(([-1,1]^n,\vert\,{\cdot}\,\vert_{\infty})\) to \((\mathbb{R}^m,\vert\,{\cdot}\,\vert)\), the \(\mathcal{H}^n\)-measure of a typical image is not bounded below by any \(\Delta>0\).