  The VLT Multi Unit Spectroscopic Explorer (MUSE) integral-field spectrograph
can detect Ly\alpha{} emitters (LAE) in the redshift range $2.8 \lesssim z
\lesssim 6.7$ in a homogeneous way. Ongoing MUSE surveys will notably probe
faint Ly\alpha{} 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 catalogs generated with a
semi-analytic model of galaxy formation coupled to numerical Ly\alpha{}
radiation transfer models in gas outflows. We expect $\approx$ 1500 bright LAEs
($F_{Ly\alpha}$ $\gtrsim$ $10^{-17}$ erg s$^{-1}$ cm$^{-2}$) in a typical
Shallow Field (SF) survey carried over $\approx$ 100 arcmin$^2$, and $\approx$
2,000 sources as faint as $10^{-18}$ erg s$^{-1}$ cm$^{-2}$ in a Medium-Deep
Field (MDF) survey over 10 arcmin$^2$. In a typical Deep Field (DF) survey of 1
arcmin$^2$, we predict that $\approx$ 500 extremely faint LAEs ($F_{Ly\alpha}$
$\gtrsim$ $4 \times 10^{-19}$ erg s$^{-1}$ cm$^{-2}$) will be found. Our
results suggest that faint Ly\alpha{} sources contribute significantly to the
cosmic Ly\alpha{} luminosity and SFR budget. While the host halos of bright
LAEs at z $\approx$ 3 and 6 have descendants with median masses of $2 \times
10^{12}$ and $5 \times 10^{13}$ $M_{\odot}$ respectively, the faintest sources
detectable by MUSE at these redshifts are predicted to reside in halos which
evolve into typical sub-$L^{*}$ and $L^{*}$ galaxy halos 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 halos at z $\approx$ 3.
