Association studies to discover links between genetic markers and phenotypes are central to bioinformatics. Methods of regularized regression, such as variants of the Lasso, are popular for this task. Despite the good predictive performance of these methods in the average case, they suffer from unstable selections of correlated variables and inconsistent selections of linearly dependent variables. Unfortunately, as we demonstrate empirically, such problematic situations of correlated and linearly dependent variables often exist in genomic datasets and lead to under-performance of classical methods of variable selection.