While Anna and William are sitting in his living room and he is reading a newspaper, she notices the size of his feet. It is clear that there is a stand-in for Will in the shots looking at Anna because his toes are different than Hugh Grant's. When you are looking from Anna's perspective, you can see that Hugh Grant's "smaller" toes are all smaller than his big toe. The stand-in's "smaller" toes are longer than his big toe.
Three separate times during the movie, the same mother and child are seen in the alley beside William's book shop. All three times this person and child are wearing the same clothes and are in the same physical position. According to the chronology of the film, they would have been stood in that same spot, not moving, for over a year.
As William first enters the elevator at the Ritz, he asks the other person what floor they're going to. They respond "three", but when they both leave the elevator, the number "1" is clearly visible on the elevator's floor indicator.
When Anna first comes into William's house, William tells her, "The bathroom's on the top floor, and the telephone's just... just up there" (pointing to the landing, one flight up). However, we see later that the bathroom's only two flights up; William's bedroom is on the top floor, three flights up. (While there's a phone in the kitchen, he presumably wants to let her have some privacy.)
When Anna wants to use the bathroom, it is the door seen from the bottom of the stairs. Later on that same day it becomes William's bedroom door. Also, in one shot, the bathroom door has windows. Later in the movie, it becomes a plan wooden door.
When Anna first comes into William's house and William tells her the way to the bathroom, the blackboard beside William says in big letters "SPIKE CLEAN UP". In the next shot from behind William the letters on the blackboard are much smaller.
When William Thacker visits the set at Hampstead Heath he is stopped by a security guard at a checkpoint. When he enters the set after speaking with Anna Scott not only is the guard gone but so is any type of security checkpoint.
In the first scene where William encounters Anna, Rufus (the book thief) is shown several times in the background. Initially he has several days growth of beard, and in one of the shots he appears to be freshly shaven, then back to the growth of beard.
Honey starts the film with shoulder length hair. In the "Ain't No Sunshine" montage, Honey's hair is cut short, understandable in the passage of time, but in a subsequent scene in the book shop when she gives Will Anna's agents' phone numbers, her hair is long again. When she announces she's getting engaged to Spike at the closing of Tony's restaurant, it is short again. The film only spans about a year.
Near the end of the movie, Anna Scott goes to the store to talk to William Thacker. When he comes out of his office and stands in front of a desk, behind him is a stack of books, all the same style. They are all facing the same way. The next scene the top two are facing another way from the bottom one.
When William leaves Hampstead Heath after overhearing Anna talking to the other actor on set, he is wearing a dark blue shirt and has no jacket. The next shot has him entering the book shop with a light blue shirt and a jacket slung over his shoulder. (The next scene, which is set the next day, has him wearing the pink shirt for the remainder of the film.)
An addled customer enters William's travel bookstore and asks for books by Charles Dickens. He's sent away with an admonished, "This is a travel bookstore", but a true travel book professional would know that Dickens wrote several notable travel books, including "American Notes" and "Pictures from Italy".
Although Bella is in a wheelchair and complains of the fact that her house needs ramps for her to get around, there is a step without a ramp leading to her front door. In one scene she is seen about to leave the house in her wheelchair, unassisted.
When William bumps into Anna with the orange juice, his arm is slightly stretched out. It would be hard to hit Anna, soak her clothes with orange juice and get such a load of the fluid on his own shirt.