Spectrum unfolding is a key tool used together with diagnostics in the determination of nuclear fields that are associated with a range of nuclear technologies spanning fusion, fission, nuclear medicine and accelerator domains. The underlying process requires a mathematical method for solving the Fredholm integral equation of the first kind. This paper discusses the development, testing and comparison of the modern combined framework of methods for performing neutron spectrum unfolding SPECTRA-UF, which includes the UF_G and UF_M subroutines, based on the underlying mathematics of the GRAVEL and MAXED methods respectively, along with a custom parameterised subroutine, UF_P. We compared the behaviour of each method using a set of synthetic data. We discuss the challenges associated with unfolding fusion spectra, and the behaviour of each subroutine along with the feasibility of using general parameterised spectra as initial a priori spectra. The UF_M, UF_G and UF_P methods showed reasonable agreement where good a priori was supplied and all improved on the a priori spectrum given, but behaved poorly where less accurate a priori was provided, with UF_G showing itself to rely more heavily on the a priori spectrum given. The UF_M subroutine performed most favourably, producing the lowest mean fractional deviation across the majority of spectra. The UF_P was able to represent the fusion peaks and relatively smooth epi-thermal regions, but performed less well where the flux spanned many orders of magnitude. The modelling of the down-scatter component of the fusion peaks was also challenging to reliably model using simple distributions.