"Very truly yours" is a more formal and dated closing. It might be appropriate in very formal settings, like official letters to government officials or highly formal business correspondence. However, in most cases, "Sincerely" is the better option.
Here's a quick comparison:
Sincerely:
* Formal: Yes
* Common: Yes
* Appropriate for most situations: Yes
Very truly yours:
* Formal: Very formal
* Common: No
* Appropriate for most situations: No
Ultimately, the best closing depends on the context of your letter. Consider your relationship with the recipient, the tone of your letter, and the industry you are in.