This paper is mainly about the areas of non-Newtonian fluid mechanics which are not investigated or not appropriately and sufficiently investigated. In fact, this should also include emerging areas of research in the field of non-Newtonian fluid mechanics due to new scientific and technological developments and advancements. The purpose of the paper is to highlight and draw the attention to these areas so that researchers (especially the young researchers and new-comers to research such as PhD students) invest their resources and efforts in these areas instead of investing in other areas which are previously investigated and hence they are of less priority from this aspect. Apart from the obvious benefit of "leveling up" in research, the attention to these rather neglected and non-explored areas of research can be beneficial at the scientific and individual levels since it can lead to breakthroughs and new discoveries in these areas of research by inspecting and assessing their potentials and impacts at the theoretical and practical levels and probing their beneficial applications. We will also provide a brief discussion about the possibility of introducing novel tools and methods in these areas of research (and in non-Newtonian fluid mechanics research in general) as well as highlighting some of the existing limitations of the past and current research in the field of non-Newtonian fluid mechanics (noting that this discussion should help in achieving the ultimate objective of this investigation).