Is the usage of 'personally' in 'I personally don't like something' redundant?

What is the difference between the following?

• I personally don't like wax museums.
• I don't like wax museums.

The adverb personally does not seem to emphasize anything here. Is it redundant?

Adding the word "personally" communicates that the speaker recognizes the subjective nature of their preference. It also precludes interpretation of the statement as an effort to persuade the listener.

I personally try to avoid using it.

It's technically redundant, since all opinions are personal, however it can modify the tone of the statement to some degree. It effectively takes emphasis away from the opinion itself by placing the emphasis on the holder of the opinion.

For example, suppose I'm in a group of people, and we're trying to decide whether to go to Madame Tussauds or to see a play. If I say, "I don't like wax museums," the tone is very final and definitive; it sounds like I'm insisting on going to the play. On the other hand, "I personally don't like wax museums," sounds like a much softer statement, and implies that while this is my opinion, I may yet be willing to go.

I think that, although the term is overused, it definitely serves a very specific purpose.

Your example is:

I personally don't like wax museums

and

I don't like wax museums

In this particular situation, by using the word "personally", the individual emphasizes that others might be of another belief.

If I ask someone, "Do you like my drawing?", and they respond with "Personally, I think it is a bit too dark.", I can tell that the person is telling me that the drawing is too dark for them given their personal tastes, but might very well not be too dark for a lot of other people. The added personally is what emphasizes the extent to which the opinion holder is unsure that others would share the same opinion.

At least that's what I personally believe.

It depends on the context and relationship of the subject and audience.

If the person typically or frequently wears the hat or shirt of some organization, and the audience recognizes that association, including the adverb 'personally' helps distinguish the speaker's opinion from the opinion of the organization.