Broken Base: Some Torchwood fans really dislike it, especially the US influence. Others feel the American characters are better-written than Jack, Gwen, and Rhys.
Complete Monster: Oswald Danes. At the end of the third episode, Jack confronts Danes about the girl he killed, claiming that he knows Danes doesn't really feel regret for it -- and Danes admits that Jack's right. Not only does he feel no regret for what he did, he claims it was the best moment of his life. He ends up working with the main cast, but they all want to kill him. Rhys is almost gleeful at the idea of killing him. Despite Jack's claims of him being a Death Seeker, and even when doing his part to save the whole of mankind from unending agony, Oswald screams out that he'll chase Suzie Cabina into hell itself, and that she'd better run faster.
And then there's the Families, who in addition to being responsible for the Fate Worse Than Death of the entire world, are also control freaks Up to Eleven, desiring nothing more than complete control over "who gets to live, for how long, and where" and willing to shoot Esther to make the decision to save the world more of a Sadistic Choice.
Jack risks capture twice to appeal to Oswald Danes' better nature in hopes that the Complete Monster will do the right thing on the word of Jack, a stranger, just because. While a first attempt might make sense just to leave no stone unturned, by the second time it should have been obvious that Oswald is on Phicorp's side for as long as it is supporting him.
When Vera and Rex each find out about the ovens in turn, their reaction is to start yelling at the middle manager in charge of one and threatening him with legal and personal consequences. This accomplishes nothing good but leads to a lot of pain and suffering for them.
Gwen's plan to get her father out of a Category 1 facility is as follows: Get a "job" as a nurse with little knowledge of medicine, find father, ???, drive out with Rhys. Needless to say, it fails.
Moral Event Horizon: Carl Malony, the guy running one of the overflow camps in episode 5. He cuts corners to save money and leaving treatable people in pain simply because they don't have insurance is bad enough, but that's only enough to classify him as a corrupt Obstructive Bureaucrat. He crosses the line by shooting Dr. Juarez to keep her from ratting him out, and then sticking her in an incinerator with the Category 1s. Then Rex comes in and gets caught, threatening to do the same if arrested, sp Maloney tries to jam a pen in his heart. Then he tries to kill Esther, but thankfully she got away thanks to his accomplice finally remembering his conscience.
Ruined FOREVER: Some fans feel that the series was drawn out, pointless, filler, Americanized, stupid, and overall Ruined FOREVER.
Ship Tease: Esther gets a load of doe eyed looks towards Rex, implications from various characters that Esther has feelings for Rex, but it remains as Subtext.
Straw Man Has a Point: Our heroes are horrified to discover that the governments are planning to warehouse "category 1s" and ultimately burn them, and go to extraordinary efforts to stop the process, but... what exactly is the correct response to the situation the world finds itself in? With an increasing number of increasingly decrepit non-dying bodies, acting as breeding grounds for disease, and the numbers of them will grow with literally no end in sight, what were the governments supposed to do? Something of this nature would have to be done eventually, there's just no way around it. (Well, unless someone stops the Miracle, of course, but there's not a shred of evidence that anyone other than our heroes has any leads on that. They only get their leads because people start trying to kill them! Nobody else would have had any starting place at all.)
It's even noted in the show - they blew these places up in episode five, in episode eight they're talking about them as every day facts of life.
Tear Jerker: When Gwen is forcefully (but well-intentionally) separated from Rhys and Anwen at Heathrow Airport.
Watching Vera get immolated. Yeesh.
Watching Gwen and her mum saying goodbye to Gwen's father after he is found hiding.
Esther's death, and Rex being forced to let her die in order to save the world.
They Changed It, Now It Sucks: Some fans don't like the fact it's become a US co-production, that they get it before the UK, and that the show itself is supposedly more "Americanized".