Read these four tutorials, which introduce four major classifications of fallacies. Although there are many possible ways of categorizing fallacies, the four major groups discussed in these tutorials are fairly standard.
Fallacies of insufficiency are cases where insufficient evidence is provided in support of a claim. Probably most common fallacies fall within this category. Here are a few popular types:
In both cases the observations are relevant to the conclusion, but a lot more data is needed to support the conclusion, e.g. Studies show that many other people who eat instant noodles live longer, and those who encounter black cats are more likely to suffer from accidents.
If someone is guilty, it would indeed be hard to find evidence showing that he is innocent. But perhaps there is no evidence to point either way, so lack of evidence is not enough to prove guilt.
Many naturalistic fallacies are examples of fallacy of insufficiency. Empirical facts by themselves are not sufficient for normative conclusions, even if they are relevant.
There are many other kinds of fallacy of insufficiency. See if you can identify some of them.