Taggers targeted the Hydro Leisure Centre off Jenny Fields Drive, covering all the fire doors around the building with their mark.
They also defaced a mobile NHS breast screening unit which was parked next to the leisure centre, a bus stop and two telephone exchange boxes next to the Hydro underpass.
It is believed the tags were painted on Friday 12 October 2012.
Officers have released images of the tags and are urging anyone who can help them to identify the offenders to come forward.
PCSO Michael Smith, of Harrogate Outer Safer Neighbourhood Team, said:
These graffiti tags are very unsightly and not very pleasant for residents and people who work in the area to look it.
It is also very inconvenient, costly and time consuming to have them removed.
The tags are quite distinctive and I am appealing to anyone who knows who is responsible to contact the police or Crimestoppers with information.
I would also appeal to anyone who witnessed any suspicious activity in the area at the time of the incidents to get in touch.
Anyone who can help officers with their enquiries is urged to contact North Yorkshire Police on 101 – select option 2 – and ask for Michael Smith or Harrogate police.
Crimestoppers can be contacted anonymously on 0800 555 111.