Quantile regression \parencite{Koenker1978} is a robust and practically useful way to efficiently model quantile varying correlation and predict varied response quantiles of interest. This article constructs and tests MM algorithms, which are simple to code and have been suggested superior to some other prominent quantile regression methods in nonregularized problems \parencite{Pietrosanu2017}, in an array of linear quantile regression settings. Simulation studies comparing MM to existing tested methods and applications to various real data sets have corroborated our algorithms' effectiveness.