The ability to delete discussions and/or comments is a permission typically reserved for Admins and Mods.
It is controlled with Roles and Permissions, and can be set on a per category level:
I deleted a post accidentally! What can I do!?
You can restore it from the change log. Please note that items deleted forever from the change log cannot be recovered.
Can users delete their own discussions and comments?
A discussion may have many comments, and deleting a discussion with many comments 'orphans' those comments, essentially deleting them, we only allow admins and mods to delete discussions.
We can enable an option in your config that will allow users to delete their own comments (this can be requested through your CSM or Vanilla Support).
How can users request to have a discussion deleted?
Some communities will create an additional option using the Reported Posts addon called 'please delete this discussion' to allow users to have an easy way to request this from the community management team so that they can assess if the discussion has many comments, and if perhaps it is preferable to edit (or blank) the discussion and change the author rather than deleting the post outright.