How do you prioritise when several issues come in at once?

Give a framework, since triage under pressure is the core operational skill.

Prioritise on impact and urgency together:

• Scope of impact. An entire site down beats one user with a broken printer, regardless of who is shouting loudest.
• Business criticality. The system that processes revenue outranks the internal wiki. Knowing which systems are critical before an incident is what makes this fast.
• Is it getting worse? A disk filling up outranks a stable failure, because the window to act is closing.
• Is there a workaround? A severe problem with a viable workaround can wait behind a moderate one with none.
• Security incidents jump the queue, because the cost of delay compounds.

The management part: delegate what others can handle, communicate expected timelines to everyone waiting so they stop chasing, and escalate early rather than when it is already too late.

Note: Mentioning that you look for the common cause when several tickets arrive together is a good detail. Five unrelated-seeming issues at once are usually one underlying failure, and recognising that saves hours.