Gardai believe criminal charges are "imminent" in two of the five incidents in which the €80,000 statue of the late Dubliners singer Luke Kelly was vandalised.
Sophisticated new security cameras and illuminating lights are to be placed around the bust due to graffiti being daubed on it five times in the 16 months since it was erected.
Last Friday night, the statue in Sheriff Street, designed by artist Vera Klute, was again sprayed with red paint in the shape of glasses and the words, "F**k Luke Kelly" were added at the foot of the bust.
A Garda spokesperson said: "An Garda Siochana is actively investigating all incidents of criminal damage to the statue of Luke Kelly.
"With reference to two of these incidents, gardai are pursuing a definite line of enquiry, with charges before the courts, imminent.
"For the outstanding cases we are still making enquiries, and we are seeking help from the public to come forward with any information, however trivial, that can assist us".
Local Independent councillors Christy Burke and Anthony Flynn voiced their disgust at the latest defacing.
The work was commissioned by Dublin City Council in 2014 when Mr Burke was mayor.
The statue is a 1.8-metre-high marble bust, with copper wire used for Kelly's trademark hair and beard. It stands at the corner of Guild Street and Sheriff Street at the mouth of the Royal Canal, close to Kelly's childhood home.
After being vandalised, it was cleaned early on Saturday morning.
Dublin City Council said it was liaising with the artist and continues to explore options with an Garda Siochana, "including direct CCTV coverage of the sculpture and enhanced lighting to reduce the chance of this vandalism occurring again".
Calls were previously made for the statue to be moved and now Justice Minister Charlie Flanagan has rowed in on wanting it relocated.
Mr Flanagan said: "We can't continue to have a situation where the monument is continuously subjected to this type of vandalism. I think all options should now be considered."
However, Mr Burke said the statue will not be moved and the council and gardai have been working on installing additional security cameras, and will be illuminating it.
"The legacy of Luke Kelly will never be removed. The gardai along with the council will be installing sophisticated security cameras and illuminating lights. The residents do not want him moved, nor do his family.
"The family know that the people around here are decent people but whoever is carrying out this thuggery are causing so much upset to residents and I assume his family.
"Shame on whoever did this. People around this area adore the statue and treat it as if it is Luke and he is still alive.
"Locals were out cleaning the area last night and they were delighted that so many visitors wanted their photos taken with the statue."
Every time the statue is cleaned, it costs €240 plus VAT.
Ruairi O Cuiv, public art manager with the council, has said that moving it from its canal basin setting may not be the best option as it is site-specific.
The statue was unveiled by President Michael D Higgins in January last year on the 35th anniversary of the 43-year-old singer's death.
A second €80,000 statue was designed and created by John Coll. It is located close to the National Conference Centre and was paid for by Gerry Hunt, Luke Kelly fan and friend, who died last year at the age of 82.
Gardai are appealing for anyone with information to contact them at Store Street on 01 666 8000 or the Garda Confidential Line on 1800 666 111.