This thesis reports on two graphene-based structures that have been proposed and fabricated as possible prototypes for high-spatial-resolution Hall sensors with potential application in research on high-density magnetic recording technology such as bit patterned magnetic recording (BPMR) and other areas where the measurement of highly inhomogeneous fields is required. There is a direct graphene-metal contact in the first structure, which is named as TYPE I in this thesis, so that the anomalous Hall effect (AHE) in the ferromagnetic islands deposited on the graphene could be detected. Meanwhile, the graphene and the metal are isolated by an h-BN layer in the second structure which is named as TYPE II, so that only the stray field from the islands can be detected using the ordinary Hall effect (OHE).The transport measurements performed on TYPE I devices revealed there is no AHE or stray field signal detectable, and their Hall resistance relations are non-linear and do not pass through the origin point. A finite element simulation comparing the resistance of the empty graphene cross and the island-occupied cross indicates that the current in the graphene may not redistribute through the metallic islands due to interface current blocking, resulting in the non-appearance of the expected AHE signal. Moreover, an analysis on the data of the longitudinal magnetoresistance (MR) reveals that a two-fluid model and effective medium theory (EMT) model might be the major graphene MR mechanisms in the regime away from and near to the charge neutrality point (CNP) respectively. As a combined result of the above findings, a joint MR-Hall effect model under the condition of the presence of a pre-existing transverse offset current, is proposed to explain the unusual behaviour of the Hall measurement data of the TYPE I devices. The model gives qualitatively correct fitting for all longitudinal and transverse transport data of TYPE I devices. In addition, the nature of the graphene/metal contact is considered as the reason responsible for the non-appearance of the expected AHE and stray field signal, although further experimental work is needed, and suggested in the thesis, to clarify this issue. On the other hand, the TYPE II devices have shown their potential to be developed as a Hall sensor being able to detect a sub-micron magnetic island in the future, but there is still a large space for the performance of the devices to be improved. At the end of the thesis, future experimental work, which could lead to the eventual development of a high-sensitivity high-spatial-resolution Hall sensor on the basis of TYPE I and TYPE II structures, are suggested and described.