Ambulances were diverted during the three-hour lockdown of University Hospital Crosshouse in Kilmarnock, which was lifted shortly before 23:00.
Police were called to the first incident at the East Ayrshire hospital at about 19:45.
They believe another in the centre of Kilmarnock about 20 minutes later and a serious crash on the A76 may be linked.
The incidents are not being treated as terror-related.
In a statement on Twitter Police Scotland said there was no cause for concern for the wider community.
It added that inquiries were at an early stage and that the areas involved have been cordoned off.
The second incident was on Portland Street, which is about two miles from the hospital, at about 20:05.
Police have not confirmed the nature of these incidents.
Officers were called to the crash on the A76, on the outskirts of the town, at about 20:30.
It was confirmed just before 23:00 that the hospital lockdown measures had been lifted.
Dr Crawford McGuffie, the medical director of NHS Ayrshire and Arran, said police had confirmed the risk to patients, staff and visitors to the hospital was minimal.
He added: "There remains a large police presence on the grounds of the hospital.
"We would like to reassure anyone coming to the hospital, in particular to any patients or staff coming on shift, that Police Scotland have confirmed it is safe to do so.
"However, if you are worried, please speak to the onsite Police Scotland officers."
First Minister Nicola Sturgeon described the incidents as very serious.
She added: "My thoughts are with those involved in these very serious incidents... Please help police do their job by following the advice to stay away from the areas affected."
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