The PIL would come up for hearing on February 10 before the bench headed by Justice P V Hardas.
Singh resigned last week, to join the politics.
The PIL refers to media reports that the new commissioner has not been appointed yet because of differences between the ruling alliance partners Congress and NCP.
If such appointments were made for political considerations, then the people will have to pay a heavy price, the PIL says.
It also seeks an affidavit from the state, stating reasons for dropping a post of Director General of Police in 2012-13 and allowing Satyapal Singh to continue as Mumbai Police Commissioner while denying him a promotion as a DGP.
The PIL also seeks a direction to the state government to submit minutes of meetings held since January 31 to decide on the commissioner and furnish list of eligible candidates to the court.