This thesis is concerned with certain types of weak singularity in general relativity for which some geometrical concepts remain well defined at the singularity. We review the use of holonomy to analyse quasi-regular singularities. We introduce a class of curvature singularities which we call idealised cosmic strings which may provide more general models for cosmic strings than quasi-regular singularities. We analyse these singularities using methods of holonomy and examine the curvature and geometry in their neighbourhoods. In order to do this we prove a number of results about the behaviour and divergence of tensors in parallelly propagated frames and in pairs of frames related by bounded transformations. Making use of path-ordered expotentials of curvature we give conditions under which we prove that certain elements of holonomy exist even for a curvature singularity. We then present a 2 + 2 formalism suited to analysing idealised cosmic strings and show how the geometry of the full connection is related to the geometry of a connection which we call the projected connection. We also apply these results to prove the existence of certain intrinsic and extrinsic holonomy groups which we define. In addition we prove a number of results about conformal singularities and in particular that the 4-cone is not conformally regular and we examine the effect of conformal transformations on extrinsic curvature. Finally we prove the coordinates may be found in which a metric has block diagonal form.