A “case study” is more formal than an anecdote, but still has the same issues.

Here’s a quote from the end of the “Limitations” section of the Wikipedia article on “Case Study”:

As small-N research should not rely on random sampling, scholars must be careful in avoiding selection bias when picking suitable cases. A common criticism of qualitative scholarship is that cases are chosen because they are consistent with the scholar’s preconceived notions, resulting in biased research.

Another quote from earlier in that section:

The authors’ recommendation is to increase the number of observations … because few observations make it harder to estimate multiple causal effects, as well as increase the risk that there is measurement error, and that an event in a single case was caused by random error or unobservable factors.

The “Uses” section of that article starts with:

Case studies have commonly been seen as a fruitful way to come up with hypotheses and generate theories. Case studies are useful for understanding outliers or deviant cases.

Lower down that section has:

Case studies of cases that defy existing theoretical expectations may contribute knowledge by delineating why the cases violate theoretical predictions and specifying the scope conditions of the theory.

Case studies are used to guide experimental and quantitative research, but are not a replacement for that part of the research process.

Applying that to case studies that appear to involve the supernatural, sufficient convincing case studies should lead to theories about the conditions for supernatural events, which should lead to experiments or quantitative studies to test those theories.