A graph G is locally irregular if every two of its adjacent vertices have distinct degrees. Recently, Baudon et al. introduced the notion of decomposition into locally irregular subgraphs, where by a decomposition we mean an edge-partition, and conjectured that almost all graphs should admit a decomposition into at most 3 locally irregular subgraphs. Though this conjecture involves a constant term, exhibiting even a non-constant such upper bound seems tough. We herein investigate the consequences on this question of allowing a decomposition to include regular components as well. As a main result, we prove that every bipartite graph admits such a decomposition into at most 6 subgraphs, this constant upper bound having no equivalent in the original context of decomposition into locally irregular components only. This result implies that every graph G admits a decomposition into at most 6 log( (G)) subgraphs whose components are regular or locally irregular.