We present an improved theoretical prediction of the positron energy spectrum for the polarised Michel decay µ⁺ → e⁺ν_eν̄_µ. In addition to the full next-to-next-to-leading order correction of order α² 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 α⁴. 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 µ⁺ → e⁺X of a muon into an axion-like particle X.