There are many advices given below but I believe if it is very pervasive, it represents a system wide failure. Think about it. If the mgt is competent, they should be able to spot it long ago. The fact they didn't, it means either they are incompetent or they asked the bully to do it. Either way, it is not ur job to repair the system.

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think about the values you stand for. If what they are doing represents your values, you should definitely stay. If what they are doing is not reflective of what you stand for, understand you don't have to be a victim of reckless mgt.

It depends on who is the bully - colleague (senior/junior in terms of hierarchy, not age) or supervisor; and form of the bully act (e.g. telling false tales about you, assigning many more tasks comparably without additional resources and with no intent of promoting you - quiet firing).

If it is a colleague of no help to your work or has no potential of being your supervisor, just find an opportunity to tell him/her off when in the company of your friendlier colleagues. Otherwise, work to prove your worth so as to be in the supervisors' good books. This bully colleague will certainly think twice then.

If the supervisor is the bully, find out how others perceive him/her first. When most said good about him/her, doubt you have a chance with HR. Try arranging a chat instead to understand his/her expectation of your work. Sometimes, it could simply be a mismatch of perceptions. If you sense during the session the bully act is biased and targeted, see whether you can make request for transfer to another department subsequently should you wish to continue working in the organisation. Otherwise, look for a new job or F.I.R.E (if you could afford to).

Usually, bully acts occur when colleague deems you as a threat at work. Only incapable people resort to detestable means. Understandably, it could be both annoying and demoralising when no aid is at sight. If work at the private, can report to MOM with loads of evidence. However, with MOM stepping in, doubt you would have a promising future after the dust settles. Anyway, don't waste precious time over such senseless workplace bully. As the chinese saying goes, 善有善报，恶有恶报；不是不报，时辰未到。What goes around, comes around. These people will get their dues eventually.

If your manager or HR cannot look into this incident. You can call the company ethics hotline if there is one.

Don't quit.