We review the theory of the Lamb shift for muonium, provide an updated numerical value and present the prospects of the Mu-MASS collaboration at PSI to improve upon their recent measurement. Due to its smaller nuclear mass, the contributions of the higher-order recoil corrections (160 kHz level) and nucleus self-energy (40 kHz level) are enhanced for muonium compared to hydrogen where those are below the level of the latest measurement performed by Hessels et al. and thus could not be tested yet. The ongoing upgrades to the Mu-MASS setup will open up the possibility to probe these contributions and improve the sensitivity of this measurement to searches for new physics in the muonic sector.