  We investigate possible effects of unparticles at the MUonE experiment by
considering a general model for unparticle with broken scale invariance,
characterized by the scaling dimension $d$ and the energy scale $\mu$ at which
the scale invariance is broken. Taking into account available relevant
constraints on the couplings of the unparticles with the Standard Model (SM)
leptons, we found that the MUonE experiment at the level of 10 ppm systematic
accuracy is sensitive to such effects if $1<d\lesssim 1.4$ and $1\le \mu
\lesssim 12$ GeV for vector unparticles. The effects of scalar unparticles are
too feeble to be detected. The vector unparticles can induce a significant
shift on the best-fit value of $a_\mu^\text{had}$ at the MUonE, thereby
providing an opportunity to detect unparticles or to obtain a new bound on the
unparticle-SM couplings in the case of no anomaly.
