Soper first made his mark in one-make series, winning the inaugural MG Metro Challenge in 1981. He made his British Saloon Car Championship debut in 1982 in an Austin Metro, and his raw pace quickly attracted the attention of team principal Tom Walkinshaw.
Walkinshaw recruited Soper to TWR for 1983, where he joined the works Austin Rover team. Soper won the BSCC on track in dominant fashion, but rival Frank Sytner lodged a protest against the TWR Rover Vitesse. The car was subsequently found to have an illegal engine installation and the team was disqualified, with the title handed to Andy Rouse. The episode remains one of the most contentious in British touring car history.
Soper went on to build his career's most enduring alliance with BMW. He joined Eggenberger Motorsport and finished as runner-up in the 1988 BSCC in a Ford Sierra RS500 Cosworth — enjoying fierce on-track battles with Rouse throughout the season — before eventually moving to a works BMW relationship that spanned multiple championships and continents.
In the BTCC, Soper was a consistent front-runner without securing the title. He challenged strongly in 1991 (finishing fourth) and was runner-up to teammate Joachim Winkelhock in the 1993 season for BMW, having led the championship for much of the year before a run of misfortune ended his challenge. He is perhaps best remembered by the British public for the 1992 season finale, in which a collision with John Cleland cost Cleland the championship and prompted Cleland to famously label Soper "an animal."
Outside the BTCC, Soper accumulated a distinguished record in endurance and international touring car racing. He won the 24 Hours Nürburgring in 1987, and claimed the 24 Hours of Spa in both 1992 and 1995. In 1995 he raced for Schnitzer BMW in the Japanese Super Touring Championship and won the title outright. He came second in the FIA GT Championship in 1997 and won the Guia Race at the Macau Grand Prix the same year.
In the 1987 World Touring Car Championship — the one-off Group A world series — Soper drove for Eggenberger Motorsport and finished first on the road at the Bathurst 1000, though the result was later annulled due to a technical infringement. He also finished fifth overall in that season.
In the German Super Tourenwagen Cup in 1996, Soper challenged for the title until a late-season incident with a teammate ended his bid; he finished second overall to Emanuele Pirro.
By 2000, Soper fell out of favour with BMW motorsport director Gerhard Berger, and retired from racing rather than compete in a different manufacturer's car. He acquired a BMW dealership in Lincoln. He made a reluctant return to the BTCC in 2001 with Peugeot, finishing sixth before being advised to retire on medical grounds following a heavy crash in the season finale. In 2013 he announced plans to return to racing after receiving medical clearance.
Soper earned the nickname "Soperman" from his fans — pronounced with the stress on the final syllable to echo "superman." His record across the BTCC, international touring car racing, and endurance events, combined with multiple expert rankings as the greatest touring car driver of his era, cements his place among the most accomplished British racing drivers of the late twentieth century.