Hi, can anyone tell me if shareholders using a custodian are given the right to either exercise or sell the rights offering given to shareholders? Or can only the custodian firm sell these rights?

When you purchase a stock through a custodian account, the purchase is not under your name but rather, the company.
IV a) 46. Where shares in a company are held through a nominee company or a custodian bank, the nominee company or custodian bank would be the registered member entitled to attend and vote at a general meeting of the company. Indirect investors such as foreign institutional investors are beneficial shareholders who hold the shares via a nominee company or custodian bank, and can only attend shareholders‘ meetings as a proxy of the nominee company or custodian bank.

https://www.acra.gov.sg/docs/default-source/def...