If you only have papers with your PhD supervisor as a co-author, a selection commision may wonder about your academic independence; whether the underlying ideas are all yours or all have been fed to you by your advisor. Having a paper without you PhD supervisor (or other senior scientists) as co-authors, helps to alleviate these worries. That this is viewed as valuable can be seen from the fact that applications for ERC starting grants specifically ask the applicant to highlight publications without there PhD supervisor. However, there are other ways of addressing/prempting such worries. It is therefore not essential to have papers without your PhD supervisor. In fact, for freshly minted PhD I think this would be perfectly normal (although how normal depends on the field).
More generally, it is good to have mixture of different types of publications. Some with, some without your PhD supervisor; some first author, some not; some single author, some in collaborations. How (and if) these are valued differs greatly from field to field, but generally it does not hurt to have the variety.
Let me close by stressing, that all these considerations are secondary to having good, impactful papers. Having your supervisor on your paper can help with both.
(PS. In the comments someone mentioned that in some fields publishing without your supervisor is seen as a cardinal sin. I'm curious about the rational of that.)