The PSNI has policed more than 70 funerals during the coronavirus pandemic.
Chief Constable Simon Byrne made the admission yesterday at a meeting of the Policing Board.
Funerals are currently restricted to a maximum of 10 mourners, but there has been criticism after photographs emerged of large crowds attending some funerals, including several republican funerals, in clear breach of social distancing rules.
Mr Bryne said the policing of funerals had been a recent "dilemma" for the force.
He added: "We have policed 76 funerals in different ways.
"We do endeavour to have a dialogue with people who are suffering loss, or the undertakers, and establish some norms and behaviours. Largely, that has resulted in success.
"There have been other occasions where people haven't abided by what we thought was an agreement. In some cases we are following up with an investigation and referrals to the prosecution service if we think offences have been committed."
When asked how many files had been submitted to the Public Prosecution Service, the PSNI said: "Investigations are continuing into a number of suspected breaches of the Health Protection (Coronavirus Restrictions) Regulations NI 2020. Once completed, files will be forwarded to the Public Prosecution Service."
The force was unavailable to provide a breakdown of how many, if any, penalty notices for disorder or community resolution notices had been issued due to breach of regulations at funerals.
Mr Bryne said policing the pandemic was a daily challenge and officers were dealing with "inconsistent events and circumstances and at the same time the environment around us is changing and the rules are changing".
He added: "We all have to try and make sense of guidelines and actual restrictions and sometimes the two overlap and become somewhat blurred.
"Those are actually some of the real-time dilemmas operational staff have to meet, even when they are stopping a car or confronting people in a park."
Separately, Mr Byrne said 730 fixed-penalty notices had been issued and 400 community resolution notices.
Last week a senior police officer told Stormont's justice committee that specific guidelines for carrying out enforcement action at funerals were never issued by the Executive.
Assistant Chief Constable Alan Todd said policing at funerals was difficult because of a lack of specific instructions in coronavirus regulations.
He told MLAs: "There's been a lot of public discourse from a whole lot of angles about funerals. What's permissible or not, what is guidance or not is a matter for the legislators and the Executive.
"Police are generally operating in a space where there is no legislation, not a lot of guidance and are trying to apply some common sense to bring about a successful outcome."
Last week First Minister Arlene Foster acknowledged that the situation was not easy.
She said the PSNI was "looking at funerals" where large numbers of people had gathered in breach of social distancing rules and that "files will be sent to the Public Prosecution Service around those issues".