Agents' Ticket Permissions dictate what actions they can take on Tickets in the helpdesk. To set individual Agent permissions, go to Agents > Agent Profiles, select the individual Agent and click on the Permission tab.
You are able to limit Agents' access to Tickets based on assignment/following statuses. Agents can always view Tickets that have been assigned to them, their Team or Tickets that they are followers of. If a Ticket is assigned to a Team, Agents in that Team will be granted the same access as if it was assigned to them directly.
If you disable View Unassigned Tickets an Agent will not be able to view any unassigned tickets in the helpdesk, if you also disabled View Tickets Assigned to Others, the Agent will only be able to view Tickets assigned to them, or one of their Teams.
Pictured below are the Ticket settings that are available for each Agent, where Agents' 'owned' Tickers refers to Tickets that are directly assigned to them.
You cannot grant an Agent permission to mark a Ticket as Spam but not permission to delete it.
This is because marking a Ticket as Spam is effectively a different way of deleting a Ticket because Spam Tickets are automatically purged after a set period of time.
To delete attachments from tickets, the 'Can Delete Messages' permission must be enabled. 
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