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The jet-in-crossflow is a critical flowfield for a range of aerospace propulsion systems, both airbreathing and rocket engines. A jet-in-crossflow or transverse jet is typically composed of a jet issuing from a circular outlet perpendicularly to a uniform air crossflow. Previous experiments deploying hot-wire anememetry or laser-based optical diagnostics such as planar laser-induced fluorescence (PLIF) or particle image velocimetry (PIV) have found that there is a transition in the jet's upstream shear layer from convective to absolute instability as the jet-to-crossflow velocity ratio $R$ or momentum flux ratio $J$ decreases. Typically, convectively unstable jets-in-crossflow are correlated with more asymmetric cross-sections while more symmetric jet cross-sections are typically associated with absolutely unstable transverse jets at lower $R$ values, usually below $3$ (Gevorkyan et al., 2016). Besnard et al. (2022) demonstrated that even low-amplitude asymmetric acoustic excitation affected the symmetry of a convectively unstable jet-in-crossflow. Such a finding suggests that there may be a linear origin to symmetry-breaking in the face of a theoretically symmetric flow field. This dissertation describes a numerical investigation of the spatial linear stability characteristics of a jet-in-crossflow for the axisymmetric and helical azimuthal modes using a 2D-local viscous linear stability analysis. This study uses the viscous hyperbolic-tangent (tanh) and uniformly valid asymptotic solution (UVAS) base flows developed by Alves et al. (2008) and Alves & Kelly (2008) as viscous extensions of the cylindrical vortex sheet solution of Coelho & Hunt (1989), who performed asymptotic expansion-based linear stability analyses of the same base flows. This present work may be considered a `fully-coupled' extension of their work because we presently account for all base flow-eigenfunction azimuthal coupling.The eigenspectra from the present fully-coupled linear stability analysis indicate that the axisymmetric mode spatially stabilises as $R$ decreases - in contradiction with experimental measurements (Megerian et al., 2007; Davitian et al., 2010; Shoji et al., 2020a), direct numerical simulation results (Iyer & Mahesh, 2016), and with the prior asymptotic expansion-based linear stability analysis of Alves et al. (2008); Alves & Kelly (2008). The wavenumber and preferred Strouhal number trends, however, are in qualitative agreement with experimental measurements. The first, second, and third helical modes all stabilise as $R$ decreases, with the axisymmetric mode being more unstable than the helical modes. All helical modes exhibit degeneracy-breaking, as was found from multiple mode analysis of an inviscid base flow by Alves et al. (2007); these modes are symmetric or anti-symmetric with respect to the plane of symmetry found in the jet-in-crossflow base flows. A novel upwind-based treatment of the linearised convective term is developed to explore features of the eigenfunctions while eliminating non-physical oscillations in the high Reynolds number regime. This consists of a hybrid central-upwind finite difference discretisation scheme suitable for convection-dominated flows that allows for surgical suppression of non-physical numerical oscillations while still yielding similar eigenvalues to purely central difference schemes. Given that only the near-wake region needed this winding, the overall scheme consists of a $4^{\mathrm{th}}$-order central finite difference for all non-convective terms, a hybrid $2^{\mathrm{nd}}$-$1^{\mathrm{st}}$-order scheme for the near-wake region and a hybrid $4^{\mathrm{th}}$-$3^{\mathrm{rd}}$-order scheme elsewhere. A linear activation function is used to smooth the transition from one regime to another. This approach was effective in both Cartesian and polar coordinates. Given that the primary difference between the present fully-coupled and prior asymptotic expansion-based linear stability analyses lies in the number of supported base flow-eigenfunction azimuthal couplings, a weakly-coupled discrete Fourier-transformed linear stability analysis is developed to allow an exploration of the effect of base flow-eigenfunction azimuthal couplings on the eigenvalues. This weakly-coupled approach can highlight the essential velocity eigenfunctions or azimuthal interactions that affect the eigenvalues the most and may be considered as a sort of reduced-order model. By reproducing the base flow-eigenfunction couplings as closely as possible without repeating the same approach, the weakly-coupled linear stability analysis yields quantitatively similar axisymmetric eigenvalues as Alves & Kelly (2008). That is to say, the UVAS base flow spatial destabilises as $R$ decreases. As the number of Fourier modes $N_f$ increases, and therefore the number of base flow-eigenfunction couplings supported, the axisymmetric mode appears to become more similar to the fully-coupled results, with stabilisation as $R$ decreases. Hence, a mechanism is proposed wherein an inadequately-expanded asymptotic base flow leads to incomplete base flow-eigenfunction couplings that deleteriously affect the eigenvalues. Without those couplings, the surviving terms may contribute to eigenvalue stabilisation. To explore this hypothesis further, a spatial kinetic energy budget analysis is developed, showing the contribution of various physical mechanisms (`Advection', `Production', and `Pressure-velocity') to the spatial growth of the kinetic energy. The `Pressure-velocity' correlation term is unique to the spatial formulation. Typically, for example, for a one-dimensional parallel Blasius boundary layer, the `Pressure-velocity' term suppresses the energy when the disturbance is destabilising and supplies the energy when the disturbance is stabilising, per Hama et al. (1980). This implies that the `Pressure-velocity' term becomes an increasing percentage of the total energy budget as the flow destabilises. Indeed, in the present analysis for the jet-in-crossflow, the axisymmetric mode from the fully-coupled analysis exhibits a decreasing contribution from the `Pressure-velocity' term, consistent with our observed stabilisation when R is decreased. In contrast, the spatial kinetic energy budget for the weakly-coupled analysis found that the ‘Pressure-velocity’ contribution increases substantially for low R, suggesting destabilisation. All mode coupling terms are negative contributions to the energy budget - supporting the proposed mechanism. Based on the present studies, it is clear that future directions for research would include obtaining a more representative jet-in-crossflow base flow by extending the tanh or UVAS base flows to higher orders of $1/R$ or from using time-averaged experimental or numerical simulation data. Such a base flow could also be used to inform non-modal stability or resolvent analyses to better match impulsively or harmonically forced jets-in-crossflow. The wavemaker (in a 2D-local sense) could also be found, which may synergise with the passive tab disturbance of the jet-in-crossflow.