Publish a paper without supervisor
Yes, you can certainly publish your own work under your own name. There should never be a question about that. You should acknowledge help you got within the paper, of course. 

There are sometimes strategic reasons for including a supervisor as a co-author even if they haven't helped, but they fail the academic honesty test at least in minor (and maybe major) ways. But sometimes it helps a paper to get published if it has a well known co-author. 

There are also some places in which institutional pressure is placed on students to include other, non participants, as authors. This is also wrong, though done. Sometimes you have to yield to such pressure, just to protect your future prospects. 

There are other situations in which it is more appropriate to include the supervisor. For example, if he/she runs a scientific lab that you depend upon to do your work, then the supervisor (and usually many others) get included as authors because of the environment and help that they provide.