As a massive fan of old-school punk rock and bass player myself, I couldn't help myself and had to chime in; Paul used a Rick on the first album for sure, and you can absolutely hear it. It really comes thru on tracks like "Remote Control," "Janie Jones," "Complete Control,"(US) "Whiteman in Hammersmith Palaise," (US) and "Garageland"--really the whole record, it has a distinct sound. Black Market Clash is a collection of B-Sides, so maybe the first half of it is recorded using a Rick---maybe up Groovy Times/Time is Tight, but I'm not 100%. Definitely "Pressure Drop" was done with a Rickenbacker. Additionally, a bunch of UK punk bass players of that time used Ricks: Bruce Foxton (the Jam), Glen Matlock (the Sex Pistols, towards the end of his run), Alvin Gibbs (the UK Subs), Tony James (Generation X), Paul Grey (Eddie and the Hotrods/the Damned) and the fellow from the Undertones. Stateside, the bass player for GG and the Jabbers used one, as did the bass player for the Dickies, Mike Roche of TSOL, the first bass player for Social D, or at least the guy who played on Mommy's Little Monster, and Jerry Only of the Misfits, but his e.q. was so "particular" that you can't really hear it outside of Static Age, and it had some overdrive/distortion. I remember reading a long time ago that he used rotosound round wounds, as opposed to Paul's flat wounds.
As a bass player myself, I keep going back and forth between Rickenbacker and Fender, and they're both great, especially Fender P bass and the Jazz bass. For reference, I play a 2000 Fender American Standard P-bass, and really want to get my hands on a Jazz bass --had one years ago, but it got stolen out of my car--and a Rickenbacker someday, but playing live and gigging, all you really need is a *good* Fender to get by. I'm honestly kinda surprised that so many people on this thread are so up on the strings, amp/cab and sound mix, and yet you can't tell a Rickenbacker from a Fender? Also, Paul absolutely played his own basslines in the studio, aside from some tracks on Sandinista, "Rock the Casbah," and Cut the Crap. I pretty much learned how to play by playing along to him, Glen Matlock and JJ Burnell of the Stranglers. Although Fenders get a lot of attention for the first few waves of punk rock bass players, Rickenbackers were right there as well, and they deserve more recognition.
P.S. - Tony James definitely used a Rickenbacker, but I believe he also played a Gibson Thunderbird. The bass player for the Newtown Neurotics also played a Rickenbacker. If you listen to all the bands I listed, and I mean REALLY listen to the bass, you'll notice the signature Rickenbacker sound in a wide breadth of applications.