Heparin should be injected into fatty tissue, not a thigh, as is depicted in the flashback.
Richard's telescope is a reflector type. In this very common type of telescope, the eyepiece is next to the end which is pointed at the subject being viewed. Light travels down the tube to a concave mirror, then is reflected back up the tube, where it is reflected by an angled mirror out the side of the tube into the eyepiece. Richard's telescope is shown the wrong way round, in that the end which should be pointed out to sea is pointed at the wall. Later Camille points the blunt (mirror) end of the scope at the beach and looks the wrong way through the finder scope attached near the eyepiece. Doing this she would see people on the beach much further away rather than closer up! Why did they choose to use a reflector when obviously nobody on the set had the faintest idea of how to use one! (It's basic!)
The autopsy report for Valerie Dupree says that she had water in her lungs. During his summation speech, DI Poole says that she asphyxiated before she was dumped in the pool. If that were true, then she would not have been breathing and would not have any water in her lungs.
When Detective Poole finds the box of expensive tea in the victim's room, he says "Camellia Sinensis if I recall correctly" - but ALL tea (except for herbal teas) is made from the leaves of the Camellia Sinensis plant. A connoisseur such as Detective Poole would have known that.
DI Poole (Miller) makes a great deal out of the fact the victim made a very expensive cup of tea, but did not drink it prior to her apparent suicide. Later, he tests unused tea leaves for poisons, however, he did not seem to bother preserving the cup of tea the victim had preferred to test for poison.