The short answer is 'both', because the use of the word midnight is heavily dependent on its context. There is no rule saying how the word is to be used, so it must be used with great care. For example:

"Bookings can only be made on Mondays, between 10 a.m. and midnight."
"A new licence runs from midnight on the first of the month."

These are both clear, but one with midnight belonging to the previous day and one with midnight belonging to the following day.

Sometimes the subject is 'all midnights', when the question does not arise:

"In this village is always darkest about 35 minutes after midnight."

In cases where the context cannot be relied upon to place a particular event, the pair of days straddling midnight can be quoted:

"The ISO week number changes at midnight Sunday/Monday."

The above problems are further compounded when '12 midnight' is used, and this wording is best avoided when referring to an event.