  We present an improved theoretical prediction of the positron energy spectrum
for the polarised Michel decay $\mu^+\to e^+ \nu_e\bar{\nu}_\mu$. In addition
to the full next-to-next-to-leading order correction of order $\alpha^2$ in the
electromagnetic coupling, we include logarithmically enhanced terms at even
higher orders. Logarithms due to collinear emission are included at
next-to-leading accuracy up to order $\alpha^4$. At the endpoint of the Michel
spectrum, soft photon emission results in large logarithms that are resummed up
to next-to-next-to-leading logarithmic accuracy. We apply our results in the
context of the MEG II and Mu3e experiments to estimate the impact of the theory
error on the branching ratio sensitivity for the lepton-flavour-violating decay
$\mu^+\to e^+ X$ of a muon into an axion-like particle $X$.
