updated 01/21/2012 AT 12:55 PM ET
•originally published 01/21/2012 AT 2:50 PM ET
French actress-singer Paradis, 39, has spent the last few days in Paris, where on Friday she shopped in the Marais district and went for a fitting at Chanel’s headquarters on Rue Cambon.
While Paradis is expected at Monday’s Paris premiere of her film Café de Flore, Depp, 48, has been keeping a low profile in Los Angeles. He was spotted sporting a fedora and driving a luxury sports car Thursday, and he showed up later that night at the Hollywood Museum, according to sources.
He’s due in New Mexico next month to start production on his new movie, The Lone Ranger.
Earlier in the week, a solo Paradis was spotted browsing in an art gallery and buying DVDs at the department store FNAC, where she picked out a box set of films by classic comedy director Billy Wilder – highlighted by The Seven-Year Itch, about a married man tempted to stray due to the charms of Marilyn Monroe.
Reps for Depp and Paradis have still not commented on the state of their relationship. As PEOPLE reports in this week’s cover story, the never-wed couple – parents to Lily-Rose, 12, and Jack, 9 – have been leading separate lives, with some sources in their circle saying they have already split.
For more on the state of the Depp-Paradis relationship, pick up this week’s PEOPLE, on newsstands now