Shirayuki is hopelessly infatuated and in love with Kinji. She shows a passion and will around him she never does in other situations. She has also fantasised about him in certain situations. Her wallpaper on her phone has been shown to be a picture of his face, and she apparently has a planner where she has already planned to have six or seven children with Kinji. She also enjoys texting him a lot and cooking for him, much to his annoyance.
Shirayuki has shown extreme hostility to any girl who seems to have an open interest in Kinji, and Aria was the first receiver of that side of herself. While she and Aria are on better terms than pretty much anyone else, Aria still remains the only girl she would outright assault if she gets close to Kinji.
An example of one of Shirayuki's open hostilities is when Aria opened her locker and found a scroll with "Thieving Cat!" in Japanese characters inside skewered on a dagger.
Similar to Aria, Shirayuki has shown open hostility towards Riko as well, but the competition they undergo is far more diverse than the simple battling that Aria and Shirayuki undergo. Rather, they have participated in bizarre contest of ability, knowledge with increasing silliness.
Much like her relationship with Aria, she's able to set aside her [one-sided] rivalry with Riko and fight together against a common foe.