The VLT/Multi Unit Spectrograph Explorer (MUSE) integral-field spectrograph can detect Lyα emitters (LAE) in the redshift range 2.8 ≲ z ≲ 6.7 in a homogeneous way. Ongoing MUSE surveys will notably probe faint Lyα sources that are usually missed by current narrow-band surveys. We provide quantitative predictions for a typical wedding-cake observing strategy with MUSE based on mock catalogues generated with a semi-analytic model of galaxy formation coupled to numerical Lyα radiation transfer models in gas outflows. We expect ≈1500 bright LAEs (F_Lyα ≳ 10^-17 erg s^-1 cm^-2) in a typical shallow field (SF) survey carried over ≈100 arcmin² , and ≈2000 sources as faint as 10^-18 erg s^-1 cm^-2 in a medium-deep field (MDF) survey over 10 arcmin² . In a typical deep field (DF) survey of 1 arcmin² , we predict that ≈500 extremely faint LAEs (F_Lyα ≳ 4 × 10^-19 erg s^-1 cm^-2) will be found. Our results suggest that faint Lyα sources contribute significantly to the cosmic Lyα luminosity and SFR budget. While the host haloes of bright LAEs at z ≈ 3 and 6 have descendants with median masses of 2 × 10¹² and 5 × 10¹³ M_⊙, respectively, the faintest sources detectable by MUSE at these redshifts are predicted to reside in haloes which evolve into typical sub-L* and L* galaxy haloes at z = 0. We expect typical DF and MDF surveys to uncover the building blocks of Milky Way-like objects, even probing the bulk of the stellar mass content of LAEs located in their progenitor haloes at z ≈ 3.