  In this paper, the three-dimensional (3D) isentropic compressible
Navier-Stokes equations with degenerate viscosities (\textbf{ICND}) is
considered in both the whole space and the periodic domain. First, for the
corresponding Cauchy problem, when shear and bulk viscosity coefficients are
both given as a constant multiple of the density's power ($\rho^\delta$ with
$0<\delta<1$), based on some elaborate analysis of this system's intrinsic
singular structures, we show that the $L^\infty$ norm of the deformation tensor
$D(u)$ and the $L^6$ norm of $\nabla \rho^{\delta-1}$ control the possible
breakdown of regular solutions with far field vacuum. This conclusion means
that if a solution with far field vacuum of the \textbf{ICND} system is
initially regular and loses its regularity at some later time, then the
formation of singularity must be caused by losing the bound of $D(u)$ or
$\nabla \rho^{\delta-1}$ as the critical time approaches. Second, under the
additional assumption that the shear and second viscosities (respectively
$\mu(\rho)$ and $\lambda(\rho)$) satisfy the BD relation
$\lambda(\rho)=2(\mu'(\rho)\rho-\mu(\rho))$, if we consider the corresponding
problem in some periodic domain and the initial density is away from the
vacuum, it can be proved that the possible breakdown of classical solutions can
be controlled only by the $L^\infty$ norm of $D(u)$. It is worth pointing out
that, except the conclusions mentioned above, another purpose of the current
paper is to show how to understand the intrinsic singular structures of the
fluid system considered now, and then how to develop the corresponding
nonlinear energy estimates in the specially designed energy space with singular
weights for the unique regular solution with finite energy.
