Nanny McDead - S1-E2
Stupidity: Mrs. Peterson tells the cops in the first interview that the phone went 'straight to voicemail', which would happen if the phone was off, and it's not the case. There's no reason why she'd lie about this particular fact. Which ties into the larger problem that the cops are looking for the missing phone but nobody tried to call it nor the phone company told them it was still on, or the plot wouldn't work.