Reigning world junior champion Lipnitskaia, 14, went head-to-head with American rival Ashley Wagner, and the Russian came out on top, sitting first with 63.55 points, 0.46 ahead of the US champion.
Russian Elizaveta Tuktamysheva sits third on 58.26, while McCorkell endured a tough day on the French ice, currently lying last in tenth with a disappointing 43.15.
Meanwhile, Jeremy Abbott leads the way in the men's competition, holding a slim lead over Japan's Takahito Mura.
US champion Abbott is sitting on 81.18 points with Mura just behind him on 76.65 and in line for his first World Cup podium finish.
Home favourite and 2007 world champion Brian Joubert sits third with 75.46.
In the pairs' competition, Russia's Yuko Kavaguti and Alexander Smirnov skated a clean short programme to go out in front with 66.78 points.
Skate Canada silver medalists Meagan Duhamel and Eric Radford are second after Duhamel fell on her solo triple lutz.
They are currently 4.50 points behind their Russian rivals with Chinese pair Cheng Peng and Hao Zhang currently in third.
And there was plenty for the home crowd to cheer in the ice dance as Nathalie Pechalat and Fabian Bourzat sit top after the short dance.
The French duo amassed 68.48 points with Italians Anna Cappellini and Luca Lanotte are more than two points behind in second, Russians Ekaterina Riazanova and Ilia Tkachenko are third.