Muir followed up her 3,000m gold by recording an emphatic victory in the 1,500m, coming home in 4mins 5.92secs – more than three seconds ahead of Poland’s Sofia Ennaoui. Ireland’s Ciara Mageean took the bronze.
Oskan-Clarke delivered a dominant performance to come home nearly half a second clear of the 800m field in 2:02.58.
The British men also had success in the 800m as Jamie Webb – a full-time science teacher – took silver. There were also silvers for Holly Bradshaw in the pole vault, Tim Duckworth in the heptathlon and the women’s 4x400m relay team.
Muir was the undoubted star for the crowd at the Emirates Arena. The Glasgow University graduate set the pace and powered away in the penultimate lap. The gap only got bigger as Muir underlined her class and secured a unique double.
“I set myself a big test this weekend,” the 25-year-old said. “There was a lot of pressure on me, but I’m so pleased to have done the job and win the double gold.
“It’s so special for me to do this on my home track. It was such a big opportunity for me, so I’m so glad I could deliver.”
Oskan-Clarke was earlier rewarded for seizing the initiative.
The 29-year-old, whose team-mate Mari Smith finished fifth, said: “I know I’m strong, but it is just about making the right moves at the right times.
“It’s hard indoors because if you don’t do that it’s too late. I wanted to be out there early and hold on for home. I knew I’d have no regrets then.”
Webb answered the sceptics who felt he could not hold down his day job and make his mark on the major stage.
The 24-year-old hit the front in the third lap, but was soon overtaken by Spain’s Alvaro De Arriba, who finished in 1:46.83. However, Webb held off Ireland’s Mark English by setting a personal best of 1:47.13.
He said: “I’ve improved every year and I was a latecomer to the sport. I only started training at 18 properly, very progressively and we’ve got a long-term view.”
Duckworth went into the day in pole position in the heptathlon, but Jorge Urena overtook him after a dominant 60m hurdles display and matched the California-based Briton’s 5m jump in the pole vault.
Urena proved a well-deserved gold medallist after powering home more than a second ahead of the field in the 1,000m, but Duckworth set a personal best to hold on to second spot.
Blackburn athlete Bradshaw recorded a jump of 4.75m to add silver to the pole vault gold she won in 2013.
Britain rounded off the championships by claiming silver in the women’s relay.
Lavial Nielsen, Zoey Clark, Amber Anning and Eilidh Doyle trailed Poland all the way, but Doyle made ground in the final leg to give the home crowd hope before finishing in 3:29.55, with the gold medallists coming home in 3:28.77.
The British men had earlier finished fifth in their relay race.
There was disappointment for world indoor champion Andy Pozzi as he lost his 60m hurdles title.
The Stratford-upon-Avon athlete posted the second-fastest time in the semi-finals, but he finished sixth in the final.
Long jumper Abigail Irozuru and triple jumper Nathan Douglas both finished seventh in their finals while Morgan Lake came ninth in the high jump.