Orkney and Shetland and the northern tip of the mainland are the areas being hit hardest by the southern fringe of the Boxing Day storm.
An amber “be prepared” weather warning is in place for the Highlands and Northern Isles on Monday, with lesser yellow warnings issued for much of the rest of Scotland where winds are reaching 50-60mph.
The latest weather front is following a similar track to Storm Barbara, which left around 25,000 homes in the north of Scotland without power on Friday.
The Met Office said wind speeds of 94mph were recorded at Sella Ness, Shetland, on Monday.
The Fair Isle had speeds of 84mph, while Kirkwall in Orkney saw gusts of 82mph.
Met Office duty forecaster Matt Roe said: “It’s pretty windy elsewhere. We’ve got around 60mph on the west coast. Elsewhere, it’s a pretty blustery day, up to around 40mph, maybe 50mph.
“We’ve got some fairly heavy showers too and these are falling as snow across central and northern parts of Scotland. There are some fairly heavy snow showers across the Highlands as well.
“It’s basically a cold windy day across Scotland with a mix of everything. There’s frequent lightning as well with some of those showers in the north.”
Travellers have been advised to check for any disruption before setting off and transport operators are being urged to contact passengers to warn them in advance of any likely problems.
On Monday, Traffic Scotland reported sporadic wintry showers on roads such as the M8, M90 and the A720 Edinburgh City Bypass.
High wind warnings were in place on bridges such as the Clackmannanshire Bridge, Forth Road Bridge, Dornoch Bridge and Erskine Bridge.
Ferry operator CalMac warned that services across the network were subject to delays or cancellation at short notice due to Storm Conor.
The Scottish Government said there has been some disruption to power supplies with more expected as Storm Conor reaches its peak. Ministers said supplies were being restored quickly.
The Scottish Environment Agency (Sepa) had four flood alerts in place on Monday for Caithness and Sutherland, Orkney, Shetland and the Western Isles.
Alerts and warnings which were in place for more southern parts of Scotland are no longer in force.
The centre of the storm is actually north of the UK, nearer Norway.
Mr Roe explained: “We’re on the southern fringe. The southern extent of the strongest winds will be moving eastwards across more northern parts of Scotland during the afternoon. The north of Scotland will be feeling the brunt of Storm Conor, particularly the Northern Isles.”
The wintry conditions come after parts of Scotland were badly disrupted by weather in the days before Christmas.
Tuesday is expected to mark an improvement in the weather and a return to more moderate conditions.
There were, however, unseasonably warm temperatures on Christmas Day, with the mercury rising to 15.1C in Dyce, Aberdeenshire.
Scotland’s Transport Minister Humza Yousaf chaired a meeting of the Scottish Government’s resilience team on Monday with representatives of the Met Office, Transport Scotland, Police Scotland and the Scottish Environment Protection Agency (Sepa).
He said: “The weather continues to present difficulties with Storm Conor passing over Scotland during Boxing Day.
“At this time of year many people will be travelling to spend time with friends and family so we would remind the public to check before they travel and delay their journey if necessary.
“Safety must be paramount and should always take priority, this is particularly the case in the north of Scotland and the islands today.”