This thesis is an investigation of the tangled vortex lines that arise in the interference of complex waves in three dimensions; they are nodal lines of the intensity where both the real and imaginary components of the wavefield cancel out, and are singularities of the complex phase about which it sweeps out a quantised total change. We investigate the behaviour of this tangle as expressed in random degenerate eigenfunctions of the 3-torus, 3-sphere and quantum harmonic oscillator as models for wave chaos, in which many randomly weighted interfering waves produce a statistically characteristic vortex ensemble. The geometrical and topological nature of these vortex tangles is examined via large scale numerical simulations of random wave fields; local geometry is recovered with sufficient precision to confirm the connection to analytical random wave models, but we also recover the (high order) torsion that appears analytically inaccessible, and quantify the different length scales along which vortex lines decorrelate. From our simulations we recover statistics also on much larger scales, confirming a fractality of individual vortices consistent with random walks but also comparing and contrasting the scaling of the full vortex ensemble with other models of filamentary tangle. The nature of the tangling itself is also investigated, geometrically where possible but in particular topologically by testing directly whether vortex curves are knotted or linked with one another. We confirm that knots and links exist, but find their statistics greatly influenced by the nature of the random wave ensemble; vortices in the 3-torus are knotted far less than might be expected from their scales of geometrical de correlation, but in the 3-sphere and harmonic oscillator exhibit more common and more complex topology. We discuss how this result relates to the construction of each system, and finish with brief discussion of some selected topological observations.