Which of the modal verbs (have to or must) is preferable in the situations like these:
You must/have to fasten your seat belt while driving?
You must/have to stop at the STOP sign.

The law says you must/have to pay the taxes.

In such situations we can refer the action to the law. And if we break it, we will be punushed, fined etc. But at the same time it can be considered as a strong order. Or must is used only with the shade of someone's personal opinion and only have to is possible here?
[url=http://www.britishcouncil.org/learnenglish-central-grammar-must-have-to.htm ]The British Council[/url] says that both are used for obligation, and 'must' is on the way to disappearing in American English. To me, all 3 of your sentences sound fine with either verb, and I doubt any native speaker would stop to consider whether the duress was internal or external.

I think you can stop worrying about fine distinctiions between must and have to when they express obligation. They are virtually identical. In the U.S. we almost always use have to in these situations, although it is certainly not wrong to use must.

Do you have to fasten your seat belt while driving?
You have to stop at the stop sign.
The law says you have to pay the taxes.

We almost never use the must of obligation in a question.

Must I go now? (Not heard very often.)
Do I have to go now? (Very commonly heard.)

Don't substitute don't have to for must not. They have different meanings. In the U.S., we often substitute an imperative for must not.

You mustn't think that I'm not thankful, for example, can be rendered as Please don't think that I'm not thankful.

We use must for obligation, however, in "official" announcements, orders, or instructions, for example, on signs:

On a highway: Right lane must exit.
In a restaurant restroom: Employees must wash their hands before returning to work.
On instructions for applying to a college: Students must apply before May 31 to be considered for a scholarship.

Note that these are basically commands that are directed impersonally to many people at once.