Why doesn't the police investigate the events at Townsend's house?
If I remember correctly, Lester Townsend tells Roger, when they meet at the U.N. building, that he is not married, or that his wife is dead.
Then, when Roger becomes the prime suspect for the murder of Townsend, why doesn't the police investigate Roger further, finds out about his drunken driving and accusations against the Townsends, which makes the local police go to Townsend's house with him and talk to a woman claiming to be Mrs Townsend? Since there is no Mrs Townsend, this would force the police to make further investigations, possibly casting doubt upon Roger's guilt. But nothing of this seems to happen.