US Edition Three callers reported a man holding a baby over a bridge, prompting officers to attend only to discover “it was man carrying a dog that doesn't like bridges”.
One woman dialled 999 after a man shouted “you’re gorgeous” at her and another called to ask police for help in suing a benefits agency as she had no money.
The force took to the social networking site in a bid to demonstrate that most of their time is spent on “social work”.
Between 5am and 5pm they had dealt with 1140 incidents and arrested 217 people.
Chief Constable Peter Fahy said he wanted to show that only a third of the incidents his team responds to are genuine crimes.
“Policing is often seen in very simple terms, with cops chasing robbers and locking them up," he said.
"However, the reality is that this accounts for only part of the work they have to deal with.
"A lot of what we do is dealing with social problems such as missing children, people with mental health problems and domestic abuse. Often these incidents can be incredibly complex and need a lot of time, resource and expertise.
"I am not saying that we shouldn't deal with these types of incidents, far from it, but what I am saying is that this work is not recognised in league tables and measurements – yet is a huge part of what we do."
Many of the calls dealt with attempted burglaries, abandoned cars and alarms activated in error.
There were reports of loose dogs, cows and horses in the roads, domestic incidents and missing teenagers, lost and found.
A large number of calls were made by babies and children playing with the telephone, others were abandoned calls made from public telephone boxes or made in error from mobile phones in pockets.
Of the more baffling, were emergency calls made to report damage to a lamp post, an abandoned television and an unclaimed bicycle.
One woman called to report somebody putting a video on her YouTube website.
The 24-hour Twitter event – the first time that a British police force has used the internet in such a way – came as police face unprecedented budget cuts, with the Government's impending comprehensive spending review next week.
But Mr Fahy insisted it was not a publicity stunt intended to soften the blow of the cuts.
"There needs to be more focus on how the public sector as a whole is working together to tackle society's issues and problems,” he said.