The Independent Office for Police Conduct (IOPC) said it had completed assessments of four referred cases and it would be commencing investigations into two of them.
One will examine how Wayne Couzens, the serving officer charged with Ms Everard’s murder, came to sustain serious injuries while in custody.
Couzens, 48, has been charged with kidnapping and killing the marketing executive, who went missing while walking home from a friend’s flat in south London on March 3.
He sustained head injuries on March 10 and March 12 and was treated in hospital on both occasions.
The other investigation will examine an “inappropriate” graphic that was allegedly shared by an officer who took part in search operations.
The Met said the graphic was reported by a number of colleagues who were “concerned by its content”.
The IOPC said an investigation into the actions of officers at a vigil for Ms Everard held in south London on Saturday was not within its remit, despite calls from the Mayor of London Sadiq Khan.
It added that it had held discussions with the Mayor’s Office for Policing and Crime (MOPAC) and Her Majesty’s Inspectorate of Constabulary and Fire and Rescue Service (HMICFRS) to consider how concerns might be included in work being undertaken by HMICFRS.
It comes as investigations into Ms Everard’s death continue in the town of Sandwich, Kent
On Monday officers combed a supermarket car park, while specialist divers were seen searching a stretch of water, and large portions of the historic town remained cordoned off with police tape.
The searches are taking place in the area after a body found hidden in woodland at Ashford was identified as that of Ms Everard.
Scotland Yard said police have routinely been searching areas of London and Kent as part of the investigation.
In south London, hundreds of people laid floral tributes for Ms Everard at the bandstand on Clapham Common throughout the day on Monday.
Later, hundreds of protesters gathered in central London to demonstrate against the handling of a vigil to remember Ms Everard on Saturday night.
The Met urged those gathered to go home and said that “enforcement action” would be taken if people did not comply.