Well, the first thing to say is that 'brain cancer' is for journalists. We need to be a lot more precise. For a start, most 'brain' cancers are metastases from primary cancers in other organs, and no-one is suggesting that mobile phones cause them.
So we are focussed on primary tumours of the brain, but even so we need to specify which type - gliomas? meningiomas? acoustic neuromas (more accurately called schwannomas)? One of the problems of cancer epidemiology is that cancers are named both by cell type (e.g. adenocarcinoma or squamous) and by location (breast, lung). A mobile phone is held next to the ear so if it is going to cause a cancer the most likely place is the auditory or acoustic nerve, or perhaps the temporal lobe of the brain. As to which cell type is most likely to become malignant, that's for the biologists to tell us.
There is a further complication - I just said 'become malignant' but in the brain even a benign tumour which keeps on growing will squash everything else and kill you. Hence 'brain tumours' not 'brain cancer'.
Unsurprisingly there have been quite a lot of studies looking at this. Mostly they are reassuring, as is this one. There is also a useful editorial. It's an expert opinion not a formal systematic review, and we wouldn't have invented systematic review if expert review were not so unreliable. Nevertheless these experts make some useful comments.
I also note that one research group consistently finds that mobile phone use is hazardous. How come they never get a negative finding? You be surprised how often this happens. I don't think it is fraud or worse; perhaps researchers are just good at picking studies which are likely to confirm their beliefs.