Hierarchy of Evidence

How much can you trust evidence? That depends on the context and the details of the evidence itself. Each case has its own criteria for what evidence (or lack thereof) is sufficiently compelling for a conclusion.

The following is a list of broad types of evidence. They are ordered from strongest to weakest by what kind of evidence would take precedence in most contexts if its strongest form was available.

Modeling — This includes math, logic, reasoning, statistics, regression, modeling, and simulation. The correctness of a model's conclusions depends critically on the accuracy and precision of its assumptions, inputs, transitions, and architecture. When these all fit the world very closely, it may justify high confidence sufficient for your purpose.
Scientific Research — It's better when a study is randomized, controlled, and replicated by multiple reputable independent groups. A study is confounded when there are credible alternative interpretations, contradictory evidence, experimenter bias (e.g. ideology of experimenters), or conflicts of interest (e.g. a funder wants a particular result).
Authorities — It's best when there's broad agreement among relevant credible authorities.
Personal Experience — This includes your own observations, feelings, intuition, and memory. These can be superior in some cases and notoriously misleading in others. It's stronger when others share your experience and you have assurances against errors of memory and interpretation.
Analogy — You compare your case to an exemplar that's similar in relevant ways.
Anecdotes — This includes rumors, testimonials, marketing, and heresay. They are sometimes accurate but are very often exaggerated, or misleading, or inaccurate.