In this thesis, we will be studying two problems with significant small-time asymptotic structures with inherent instabilities and discontinuities. First, we consider the concentration of particles in suspension in a shear flow near a plane-wall, in which an inherent particle-free layer close to the wall forms spontaneously from a uniform concentration. We will show how the structure of this particle-free layer can be predicted at small times using matched asymptotic expansions, and the subsequent evolution of the concentration, evaluated numerically by integrating from small times. By using the calculated asymptotic structure, we can avoid the inherent discontinuity associated with the particle-free layer at t = 0. Secondly, we will be examining an inherent instability present at small times in the Kuramoto Sivashinsky equation, formulated as a perturbation of the kinematic-wave equation. This instability is invisible to regular matched asymptotic expansions, which, at small times, see no deviation in the leading-order solution from the kinematic-wave solution. However, by considering exponentially small terms in the asymptotic expansion, we can predict an asymptotic breakdown at small times, long before the kinematic-wave solution develops a shock. These terms can be found by analyzing the asymptotic structure of the solution in the complex x-plane, in which various Stokes and anti Stokes lines responsible can be found. This has consequences for, inter alia, the Navier-Stokes equations at high Reynolds number. There is evidence to suggest that, under certain circumstances, unsteady boundary-layer theory does not pro vide solutions, in the limit of high Reynolds number, to the Navier-Stokes equations. For instance, numerical calculations by Brinckman and Walker (2001) show that at sufficiently high Reynolds number, a rapid oscillation can develop in the solution, even before a breakdown occurs due to separation of the fluid. Analyzing the complex-plane structure of solutions to the Navier-Stokes equations is difficult due to the number of spatial dimensions, so instead, we consider the Kuramoto-Sivashinsky equation. This equation shares certain key properties of the Navier-Stokes equation predicted by Cowley (2001) to be responsible for the instability. It possesses a destabilizing term for a range of high wavenumbers, and it is non-linear, allowing these destabilizing modes in the Fourier spectrum to be filled out, even if no such modes are present in the initial condition. For this reason, the instability is considered inherent to the equation, which would suggest a similar instability can arise in the Navier-Stokes equation, without the need for a specific forcing or choice of initial data.