A numerical method based on an adaptive octree space discretization for the simulation of 3D free-surface fluid flows is proposed. The Navier-Stokes equations are solved with a time-splitting scheme, which decouples advection from diffusion/incompressibility. The advection step is solved with a semi-Lagrangian VOF-based scheme on the octree. An interface prediction algorithm is used to refine the octree at the predicted location of the interface in order to ensure detail preservation. Subsequently, the fluid is advected and a coarsening algorithm adapts the mesh to avoid excess refinement in non-interfacial regions. SLIC and decompression algorithms are used for post-processing to limit numerical diffusion and correct numerical compression of the VOF function. The octree scheme allows anisotropy, refinement of interfacial cells to an arbitrary level and supports arbitrary complex domains. It does not require a 2:1 cell size ratio condition between adjacent cells. The octree is then coupled with a tetrahedral mesh on which we solve the second step of the splitting algorithm, the Stokes' equations. Numerical validation is done on both advection benchmark test cases and results are compared with the uniform cell grid scheme. Paddle-generated water waves are also simulated and results are compared with experimental water wave profile measurements. \bigskip First order finite element stabilization schemes for the time-dependent Stokes' equations are studied. A unified proof of stability and convergence of velocity and pressure for consistent and non-consistent PSPG schemes for the time-dependent Stokes' equations is given with explicit dependence on viscosity and stabilization parameter. The link between bubble enrichment and Pressure Stabilized Petrov-Galerkin (PSPG) schemes in the context of time-dependent Stokes' equations is discussed and two bubble-based PSPG-type schemes are studied. Different possibilities for stabilization parameters are discussed. Numerical comparisons are done to determine stability, convergence and conditioning issues associated with different PSPG schemes, bubble-based schemes and local pressure projection schemes in different settings.