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We investigate the effect of a fixed forbidden clique minor upon the strong chromatic index, both in multigraphs and in simple graphs. We conjecture for each $k\ge 4$ that any $K_k$-minor-free multigraph of maximum degree $\Delta$ has strong chromatic index at most $\frac32(k-2)\Delta$. We present a construction certifying that if true the conjecture is asymptotically sharp as $\Delta\to\infty$. In support of the conjecture, we show it in the case $k=4$ and prove the statement for strong clique number in place of strong chromatic index. By contrast, we make a basic observation that for $K_k$-minor-free simple graphs, the problem of strong edge-colouring is "between" Hadwiger's Conjecture and its fractional relaxation. For $k\geq5$, we also show that $K_k$-minor-free multigraphs of edge-diameter at most $2$ have strong clique number at most $(k-\frac{1}{2})\Delta$., Comment: 23 pages, 4 figures; v2 includes minor corrections, to appear in Journal of Graph Theory