  Inf-sup stable FEM applied to time-dependent incompressible Navier-Stokes
flows are considered. The focus lies on robust estimates for the kinetic and
dissipation energies in a twofold sense. Firstly, pressure-robustness ensures
the fulfilment of a fundamental invariance principle and velocity error
estimates are not corrupted by the pressure approximability. Secondly,
$Re$-semi-robustness means that constants appearing on the right-hand side of
kinetic and dissipation energy error estimates (including Gronwall constants)
do not explicitly depend on the Reynolds number. Such estimates rely on the
essential regularity assumption $\nabla u \in L^1(0,T;L^\infty(\Omega))$ which
is discussed in detail. In the sense of best practice, we review and establish
pressure- and $Re$-semi-robust estimates for pointwise divergence-free
$H^1$-conforming FEM (like Scott-Vogelius pairs or certain isogeometric based
FEM) and pointwise divergence-free $H$(div)-conforming discontinuous Galerkin
FEM. For convection-dominated problems, the latter naturally includes an upwind
stabilisation for the velocity which is not gradient-based.
