For the compressible Navier-Stokes equations, a convenient possibility is to choose the time step by considering the stability criterion of the inviscid (hyperbolic, Euler) and the viscous (parabolic, diffusive) terms independently from one another:
For the <PERSON> term you have "something like"
$ \Delta t_\mathrm{Hyperbolic} \leq \displaystyle\min_{\mathbf{x}i\in\Omega} \left( \min{d=1,2,3} \frac{C \Delta x_{i,d}}{\lvert u_{i,d} \rvert + a_i} \right ) \; ,$
depending on your grid, spatial discretization and temporal discretization schemes, where $u_{i,d}$ is the d-th velocity component in the i-th cell/element, $a_i$ is the speed of sound, $\Delta x_i$ is the i-th cell/element size, and $C$ is the Courant number. Remember that in 1D the eigenvalues of the flux Jacobian for the inviscid term are $u + a$, $u - a$, and $a$ (and that you can compute them at every time step for every cell/element).
For the diffusive term you will have "something like"
$ \Delta t_\mathrm{Parabolic} \leq \displaystyle\min_{\mathbf{x}i\in\Omega} \left( \min{d=1,2,3} \frac{ \Delta x_{i,d}^2}{\lambda \; \mu} \right ) \; ,$
where $\mu$ is your diffusivity and $\lambda$ is a factor that depends on your spatial and time discretization schemes. Although it seems counterintuitive, explicit time integration schemes for the linear diffusion equation satisfying this criterion are stable for a sufficiently large $\lambda$. Implicit schemes allow you to take a larger $\Delta t_\mathrm{Parabolic}$ from the stability point-of-view, but if it is not small enough your scheme won't be very accurate.
So one could choose $\Delta t = \min ( \Delta t_{\mathrm{Hyperbolic}}, \Delta t_{\mathrm{Parabolic}} )$ and hope to remain stable for a sufficiently large class of problems if $C$ and $\lambda$ are choosen appropiately (they depend on the spatial and temporal discretization schemes!).
However, don't forget that: both terms are actually coupled, and as <PERSON> said above, the CFL condition is a necessary condition (which is not the same as sufficient!). If you are using a first order finite volume method, a consistent and monotone numerical flux ensures that your scheme is both stable and convergent.
We know from <PERSON> theorem that no linear monotone scheme of higher than first order accuracy exist. 649
This means that a higher order scheme has to be non-linear in order to be monotone. Total Variation Diminishing (TVD) methods can be constructed through limiting in FV, limiting and filtering in DG, and non-linear stabilization in FE. For higher than second order accuracy, it is worth relaxing the TVD constraint in order to retain the global order of accuracy of the method when discontinuities are present in the solution. Alternatives are Total Variation Bounded (TVB) and Total Variation Bounded in the Mean (TVBM). These methods do not remove oscillations completely, but kind of ensure stability while maintaining your global order of accuracy.
So, if you satisfy both the CFL condition and some non-linear stability condition (TVD,TVB,TVBM) your scheme will be very likely to remain stable. If you do not, you are on your own, which does not mean that your scheme will be unstable (see for example second-order FV schemes for the Euler equations using unlimited weighted least-squares reconstruction). 232