Is it usual to ask for several references for postdoc positions? I have seen postdoc job adverts which were asking for as many as three recommendation letters, despite being supervised positions and inviting freshly graduated PhD students.  Is this practice common and reasonable?

As a more practical question, how should an applicant deal with this situation?

It seems to me that it is not unusual that the only person a student worked with extensively is their advisor. In fact it would even seem like a good thing to have an advisor who does not lead a large group and has time to work closely with all their students. Yet such students would find it rather difficult to obtain three recommendation letters.
In my experience in Mathematics, three letters is very common.

So who to ask for these three letters? The better your letter writers know you and your work, the more meaningful their letters can be. But this does not mean that you must have worked closely for an extended period of time with them. Perhaps there are others that are familiar with your work, for example because they are on your thesis committee, or they have invited you to give a talk. Depending on the types of positions you are applying for, you can also have a letter that focuses on your teaching skills.

Finally, an advisor that has plenty of time to work with you, does not preclude you from also working with others. In fact, the value of building a network of colleagues cannot be underestimated. This does not necessarily mean collaboration with others, but it is important that you get out there so that your peers become aware of you and your results. These are the people that eventually may have to decide to hire you or not, or could write a letter for you.