Experimental and numerical results on premixed flame penetration and subsequent propagation in a multichannel burner are presented. The burner consists of the set of planar straight quartz channels which transverse sizes can be varied. It is found that, depending on mixture flow rate, equivalence ratio and channels transverse sizes a big variety of combustion regimes can be observed. These regimes include burner-stabilized flames, upstream propagating flames, and flames stabilized under the burner external surface. The placement of different combustion regimes in equivalence ratio/flow rate plane is plotted by means of experimental and numerical studies. In wide range of parameters, the flame pulsations consisting of repetitive stages of flame ignition, upstream propagation, and quenching take place. Results of numerical simulations obtained in the framework of simplified thermal-diffusion model are found to be in a good qualitative agreement with experimental data and allow to explain experimental findings.