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We calculate the CMB $\mu$-distortion and the angular power spectrum of itscross-correlation with the temperature anisotropy in the presence of thenon-Gaussian neutrino isocurvature density (NID) mode. While the pure GaussianNID perturbations give merely subdominant contribution to $<\mu>$ and vanishing$< \mu T>$, the latter quantity can be large enough to be detected in thefuture when the NID perturbations $\mathcal S(\mathbf x)$ are proportional tothe square of a Gaussian field $g(\mathbf x)$, i.e. $\mathcal S({\mathbfx})\propto g^2({\mathbf x})$. In particular, large $< \mu T>$ can be realizedsince Gaussian-squared perturbations can yield a relatively large bispectrum,satisfying the constraints from the power spectrum of CMB anisotropies, i.e.$\mathcal{P}_\mathcal{SS}(k_0) \sim\mathcal{P}_g^2(k_0)\lesssim10^{-10}$ at$k_0=0.05$ Mpc$^{-1}$. We also forecast constraints from the CMB temperatureand E-mode polarisation bispectra, and show that$\mathcal{P}_g(k_0)\lesssim10^{-5}$ would be allowed from Planck data. We findthat $< \mu >$ and $|l(l+1)C^{\mu T}_l|$ can respectively be as large as$10^{-9}$ and $10^{-14}$ with uncorrelated scale-invariant NID perturbationsfor $\mathcal{P}_g(k_0)=10^{-5}$. When the spectrum of the Gaussian field isblue-tilted (with spectral index $n_g \simeq 1.5$), $< \mu T>$ can be enhancedby an order of magnitude.

Anisotropic CMB distortions from non-Gaussian isocurvature perturbations