Which role is a good fit for Business Analyst in the Agile environment?

BA can work more closely with PO.
In few organizations, BA is the mediator between team and PO when PO is not approachable all the time.

Pench -

The most common models I have seen are the BA being a part of the Dev team or the BA proxying for the PO as Sonali indicated.

Kiron

I have tried BA in all three roles and here is my experience

BA as PO proxy - Most of the times POs are not available to the team and hence BA is best suited for this role. This has been the case in most projects I have taken up.

BA as Scrum Master - I have found that BA as scrum master are best suited since being neutral comes to them naturally. Have tried this few times and wasn't disappointed.

BA as Team Member - This has been the case when PO has been available and in this case BA becomes redundant during DSMs. So this has been the least productive role for BA in my experience.

So BA as PO proxy and as Scrum Master has worked well for me. Note that I work for a small organisation where usually PM or Team Lead have to take up Scrum Master role.