Summary: This paper deals with a particular issue of computational mechanics in main FEM codes nowadays available, i.e. the outcomes of implementations of large strain constitutive models based on the adoption of so-called objective stress rates, in order to satisfy objectivity requirements. The point here is that of directly inquiring whether well-known incoherencies due to the adoption of the Zaremba-Jaumann objective stress rate may manifest themselves when the most used elastic and elastoplastic constitutive models are adopted. The present investigation aims at providing a comprehensive review of the theoretical aspects and at developing an informed knowledge to final users of FEM codes, in terms of exposing which constitutive models and FEM implementations may be affected by Zaremba-Jaumann objective stress rate induced incoherencies. Towards this end, local FEM simple shear tests are explored and clearly show that kinematic cases characterized by a non zero spin may be heavily affected by oscillatory incoherencies, which arise for expected cases, i.e. Cauchy stress responses, but also for other less expected cases, i.e. strain responses, whether they are total, elastic or plastic. Beyond local tests, structural simple shear tests are also performed and show as well that oscillatory incoherencies found in local simple shear tests may heavily influence the overall structural outcomes. A non-secondary target of the paper is that of reviewing the relevant scientific and technical literature about objective stress rates, by critically analyzing correlated issues and proposed solutions, considering scientific contributions spanning over a century, keeping specific attention to the treatment of the Zaremba-Jaumann objective stress rate and to the possible flaws related to its adoption.