Socha was first recognised by agents and casting Directors when she attended a local drama workshop, having been inspired by her older brother. This led to her being cast as the lead in the Arctic Monkeys' video for their 2006 single "When the Sun Goes Down" at the age of 15.
In 2009 Socha went on to star in Samantha Morton's directorial debut The Unloved in which she played a sixteen-year-old girl in a care home. She was nominated for a BAFTA for Best Supporting Actress for her role. A week later she auditioned for the role of Kelly Bailey in British comedy-drama Misfits and got the part.
In 2010 Socha starred alongside Tim McInnerny in the short film Missing. She also had a minor role in the television mini-series Five Daughters, based on the Ipswich serial murders.
In May 2011 she won a BAFTA in the Best Supporting Actress category for her role of Kelly in Misfits. Socha was nominated for Outstanding Actress in the fiction category at the Monte-Carlo Television Festival in 2011. She later appeared in the short The Child, playing a version of Marilyn Monroe. She also took part in the BBC learning project "Off By Heart Shakespeare" where she played Juliet from Romeo and Juliet, performing the speech: "Blistered be thy tongue".
On 23 January 2012, it was reported that Socha had been charged with racially aggravated assault in connection with an incident involving a Derby taxi driver, Sakandar Iqbal, the previous October. Iqbal stated that "She called me a Paki, a dirty bastard, and said, 'You're Asian, fuck off back to where you came from'. She said, 'Do you know who I am? I'll have your family lifted'." On 2 May 2012, Socha pleaded guilty to the charge, and was sentenced to four months in prison, suspended for 12 months.
In October 2015, she announced on her Twitter that she and her boyfriend were expecting their first child together.
In April 2016, it was announced via Twitter that she had given birth to a girl, Reenie-Rae.