What is the best software for solving and simulating the 3D Navier-Stokes equation for incompressible laminar non-Newtonian fluid flow?
3D Navier-Stokes problems are already difficult to analyze, not to mention rheological concerns. I could have suggested Polyflow of Ansys but I will not because I believe it is a poorly executed idea. GUI is not good, solver options are tricky to adjust. Actually, I suggest not to rely on commercial software. Solvers in this area are not that general, they are mostly developed for specific industries, like Moldflow.
Instead, I would go with OpenFOAM. The number of non-newtonian models might not be enough for you but you still have the opportunity to adapt your own model since the code is freely available. Also, there are some efforts to implement some advanced problems, an example is in viscoelasticity.
If you can use finite-elements for space discretization and if your models are given explicitly, I suggest you use FeNiCS
The FEniCS Project is a collection of free software with an extensive list of features for automated, efficient solution of differential equations.
(says the website)
I know FeNiCS as a powerful and easy to use but clean and very mathematical software suite. In addition, it is free and open source. Maybe there are already implementations of certain models for non-Newtonian flow.
A reference for finite-elements for Non-Newtonian flows you can find here