Clara and its surprisingly not even close.
Rose was an awful, schmoopy character yes, but at least there wasn't the awful ideology behind Rose.
Clara taking centre stage was the start of Mofftwat's pandering. Basically the coward was too afraid to have a male Doctor be centre stage, so he had to emasculate the Doctor. As Iank said they basically chopped his balls off metaphorically first.
Furthermore Clara infected the classic era too. At least the crappiness of Rose was contained to the RTD era, but my god Clara had to be the one who told him what TARDIS to steal, the one who made him overcome his fear as a child, the one who saved him in every story etc.
Also Clara was genuinely nasty to the Doctor and had no chemistry with him. As awful as Rose was, at least she and Tennant did genuinely seem in love. Tennant seemed like a dashing hero, in love with his brave girlfriend, whilst the 12th Doctor seemed like a pathetic idiot, desperate to have this woman who didn't even like him hang around.
She'd hit him, she'd make jokes about him looking too old and infirm, he'd sheepishly make remarks about how he should pay her to hang around with him, and what's worse he actually had to BEG her to look after him.
Compare that to Rose who in contrast isn't sure when her Doctor changes into a younger, more handsome man and you can see who is more likable. Rose clearly likes him for who he is, hence why when he becomes a pretty boy she isn't convinced until she sees him save the day, whilst Clara turns away as soon as the young pretty boy is gone
Also Clara was more actively annoying. By that stage Moffat's writing style really began to piss me off. All of his characters explaining how genius the plot was, characters using the words clever and flirting about 5000 times each episode, added to that she was used as part of Mofftwats smutty little fantasies, like Missy and Vastra wanting to shag her.
Furthermore Clara's story arc was bollocks. Missy brought them together first because Clara was a bit of a nag, and then later because she felt that Clara and the Doctor would love each other so much he'd destroy all of reality for her?
Finally Rose also had more chemistry too. She and Tennant were a natural fit, as annoying as they were. Clara and Capaldi however had no chemistry in anyway.