We propose a detailed study of the motional dynamics of Mu and the low temperature Mu0-like states in anatase and rutile Titanium Dioxide (TiO2). An initial study on only rutile TiO2 has been completed where low TF measurements have found a Mu0-like state where the muon is coupled to an electron localized on a nearby neighbouring Ti3+ ion where two inequivalent Mu bonding configurations are measured. Significant differences in the rutile versus anatase structure open questions about the local environment in each distinct phase. A complete characterization of motional dynamics associated with this state is essential for progress in understanding the true Mu properties and behaviour. This study will provide essential data relating to the H impurity for any technological application and supplement the ongoing study of Mu excitonic states and behaviour via IR Illumination in rutile TiO2.