The man, identified as Cherif C. by French authorities who have not given his full name, lived in the Poteries area of Strasbourg about a 20-minute tram ride west of the centre of the French city and its Christmas market.
"It's a building for desperate people. No one wants to live there," one local, Bemba N'diaye, 37, told an AFP reporter near the nine-storey concrete bloc where Cherif's name is on a letterbox.
"People there are very isolated," N'diaye said.
Others said the man was known in the area owing to his criminal record, but he kept a low profile.
"His family has lived around here for a while, but he lived on his own nearby," Zach, a 22-year-old, said. "He was discreet, not a thug."
The undated photo, provided by the French police, shows Cherif C., who police are searching for in connection with the the Christmas market in Strasbourg. Photo: DPA
The suspected gunman has been sentenced an extraordinary 27 times, mostly in France where he was born, but also in Germany and Switzerland which are easily reached from Strasbourg.
His crimes range from violence to robbery, but not terrorism.
Cherif was added to a watchlist of possible extremists while in prison in France in 2015 after he "called for practising a radical form of religion," French deputy interior minister Laurent Nunez said on Wednesday.
He has since been monitored by France's domestic intelligence agency, the DGSI, which is highly stretched due to the large number of suspected extremists in France.
Some 25,000 people are currently on the "S" extremism watchlist, 9,700 of them for radicalism "linked mainly to Islamist terror movements," according to the interior ministry.
"He is an individual who has unfortunately been known for a very long time for crime," Interior Minister Christophe Castaner told French MPs on Wednesday.
"From the age of 10, his behaviour was already criminal. He had his first sentence at 13 years old," Castaner said.
On his identity card, seen by an AFP reporter, the suspect has dark eyes, black hair and a short beard.
Police had tried to detain Cherif on Tuesday morning in connection with an attempted murder enquiry, but he was not at home, Nunez told France Inter radio on Wednesday.
A grenade, four knives and a rifle were found during a search of his apartment, prosecutors have said.
He is "known for a number of criminal offences (...) but has never been linked to terrorist offences," Nunez added.
On Wednesday onlookers placed candles and messages near the scene of the crimes in Strasbourg, as the hunt for the perpetrator continued.
German authorities were on the lookout for the fugitive on Wednesday "along the Rhine" river which serves as the border between France and Germany, a spokesman from the Baden-Wuerttemberg region said.
"But at the moment we do not believe that he has crossed into the country," he added.
According to Germany's Tagesspiegel newspaper, Cherif broke into a dentist practice in Mainz, Rhineland Palatinate state, in 2012, making away with cash, stamps and gold used for teeth fillings.
Four years later, he targeted a pharmacy in the Lake Constance town of Engen, Baden-Wuerttemberg, pocketing cash. He served a year behind bars in 2016 before being expelled back to France.