There are a number of traditional beliefs about survival after death and while I doubt there's any survival in any real sense without continued bodily existence a couple of the older ideas seem to be within the realm of remote possibility because they can be investigated scientifically.
I'll admit that this was brought to mind by qewl's forum thread on ghosts, but it is something I thought about some time ago.
Most official traditional beliefs are based on someone's "vision" or hallucination involving a visit to the afterlife. There are problems with this even though such visions have similar elements even when from different traditions. Nearly all of them are from hundreds if not thousands of years in the past, making any questioning of the visionary impossible (not to say dangerous in some places). This means that while one can compare accounts of different visionaries scientifically one cannot directly investigate the content of the visions. It's still outside our scope to directly share someone else's subjective experience, let alone that of someone long dead. External comparison of literary accounts is the most we can do. Further, many such accounts, even if the original person was sincere, are edited and changed for the purpose of supporting some authority.
Spiritualism, contacting the dead through mediums, has been thoroughly investigated and exposed. Professional magicians can tell how many of their illusions are accomplished. I've seen mediums and imitated them with some success (long ago and not with any purpose but to see if I could).
Ghosts actually are remotely possible simply because different individuals can investigate them objectively, not relying on another's subjective account as "proof." This is what's been done with mediums, of course, except that one needn't be or pretend to be a medium to investigate ghosts. Haunted places can be wired with instruments and witnesses can await the haunting. I'm not claiming there are ghosts or even that if there are they are really disembodied dead people, I'm just saying that the ability to investigate them, try to measure what might happen, to duplicate others' sightings, and to publish the results places them in the realm of possible. Maybe 1% possible.
The same, oddly, goes for reincarnation. A person's claim of past lives can in theory at least be investigated objectively. Someone in a hypnotic state says his name is Joe Blow and he's an insurance salesman in Omaha in 1932 and dies of a heart attack at age 68 in 1944. One can go to the Omaha library and look in old phone books and city directories for Joe Blow and go through graveyards seeking his marker. Of course it would be more credible if a 9 year old girl in Ulan Bator made the claim. Modern records make it more possible to investigate such claims, though there are people within living memory who left almost no record: my father had a brother who died at age 12 in 1918; only family histories have any record of him because he lived in rural east Texas and his grandfather and uncles were ministers who must've seen to his funeral, the doctor's name hasn't come down to me, and his grave marker as far as I could tell has disappeared. Still, if centuries go by in N. America without serious upheaval there should be many ever more precise records (also useful for fraud).
Reincarnation can be investigated objectively according to the same principles as ghosts because external records now exist of most people that are accessible public records. Again, that doesn't mean I'm asserting it happens or could guess why it would, only that the fact it can be checked by others, the checks duplicated and published, and examined by anyone else.
Incidentally, if either of these could be shown to actually happen they would not in my opinion be supernatural but rather would be natural phenomenon.