The Humberside force sent a series of tweets about the icy roads in the county and neighbouring and North Lincolnshire.
A spokesman tweeted: “Motorists please take extra care today. The freezing overnight temperatures have left us with very icy roads and they still haven’t thawed.
“More than 30 collisions across the force area this morning. Emergency services are stretched. Please drive to the conditions.
“Rural areas across North Lincolnshire and East Yorkshire are especially icy. What look like puddles on the road are actually black ice.”
In North Yorkshire, the fire service tweeted a picture of an overturned car at Gunnerside, near Muker. The air ambulance was called out to rescue a family of five, who suffered minor injuries.
A “yellow” weather warning for ice has been issued for Scotland, Northern Ireland, the North and the East Midlands.
Icy patches are likely to form on roads and pavements as temperatures dip below freezing on Monday.
Showers of rain, sleet and snow are expected and higher ground in the affected areas could accumulate between 2-5cm of snow.
The warning extends until 11am on Monday, with people being warned they could experience “difficult driving conditions and slippery pavements”.
Met Office forecaster Emma Boorman said it was a cold start to the year that could see large parts of the UK wake up to frost.
Temperatures in parts of Scotland and the North could “quite easily” reach as low as minus 4 or 5C (23-25F), she said, due to a large area of high pressure allowing northerly winds to bring cold air into the UK.
Warning motorists to be aware of icy spots, she said: “Take extra caution on the roads, leave extra time. There could be a few icy patches on untreated surfaces, and the showers could be quite nasty too.
“If you do run into a shower, particularly in the northern third of Scotland, there could be a wintry element of sleet and snow particularly up on the hills.
“Because the breeze is coming in from the north, we tend to see the showers mostly affecting the coasts of Scotland, England and Northern Ireland,” she added.
Towards the south of the country, she said: “There’s been an awful lot of cloud and rain through the course of the day so overnight that’s going to be finally making its clearance and the temperatures will be falling away accordingly, so not quite as cold as it will be further north, but still a chilly night.”
Large swathes of England, Wales, Scotland and Northern Ireland could wake up to a light frost on Monday morning, she added.
The South East was the only place that could be immune because it was the area which had held on to the cloud and rain the longest on Sunday, with temperatures of around 2-3C (36-37F).