Monte Carlo simulations are employed to investigate the adsorption of a flexible linear cationic polyelectrolyte onto a fluid mixed membrane containing neutral (phosphatidyl-choline, PC), multivalent (phosphatidylinositol, PIP2), and monovalent (phosphatidylserine, PS) anionic lipids. We systematically study the effect of chain length and charge density of the polyelectrolyte, the solution ionic strength, as well as the membrane compositions on the conformational and interfacial properties of the model system. In particular, we explore (i) the adsorption/desorption limit, (ii) the interfacial structure variations of the adsorbing polyelectrolyte and the lipid membrane, and (iii) the overcharging of the membrane. Polyelectrolyte adsorbs on the membrane when anionic lipid demixing entropy loss and polyelectrolyte flexibility loss due to adsorption are dominated by electrostatic attraction between polyelectrolyte and anionic lipids on the membrane. Polyelectrolytes with longer chain length and higher charge density can adsorb on the membrane with increased anionic lipid density under a higher critical ionic concentration. Below the critical ionic concentration, the adsorption extent increases with the charge density and chain length of the polyelectrolyte and decreases with the ionic strength of the solution. The diffusing anionic lipids prohibit the polyelectrolyte chain from forming too long tails. The adsorbing polyelectrolyte with long chain length and high charge density can overcharge a membrane with low charge density, and conversely, the membrane charge inversion forces the polyelectrolyte chain to form extended loops and tails in the solution.