When Kitty leaves the nanny's apartment, she opens her umbrella, dislodging a fine spray of droplets from the vanes.
As she had been in the apartment long enough to feed her son, her umbrella would have either dried or been standing long enough for the rainwater to have run down to the ferrule.
At the graveside, Kitty comments that the ground is so cold and hard it would have taken nearly an hour to dig up, even with the back hoe. But the victim had only been buried the day before. The soil would have still been loose. And if it really was that cold, he wouldn't have been buried until the first thaw.
Although Holmes's brownstone is supposed to be on Stamford Ave. in Brooklyn; when walking to it he, Watson and Kitty pass Stamford Ave, one block before they get to the street where his house is (they are actually walking north on Lenox Ave/Malcolm X Blvd. to West 121 St. in Manhattan) In fact, there is no Stamford Ave. in Brooklyn or Manhattan, they film exteriors at 13 West 121 St. in Manhattan--all very ironic given the name of this episode.
After investigating the backhoe, Sherlock refers to the hair left by the perpetrator as being "dyed" which is incorrect. While using the word dye as a verb is used most often, especially by Americans, the correct verb is to colour one's hair. Dye is the product used in the colouring process. Sherlock, a language snob, should be using the correct word.