2009-10: Tommy Veilleux played for the league champion St-Joseph bantam AA team in Quebec.
2010-11: Veilleux played in nine games for the Levis Commandeurs in Quebec Midget AAA hockey – preferring to spend most of the season skating alongside his twin brother Steven, a goalie, on the Beauce-Amiante Elites Midget Espoir team. He had 1 assist and 14 penalty minutes with Levis. He was one of the top scorers for the Elites and played in the Midget Espoir all-star game. Veilleux was selected by Victoriaville in the first round (14th overall) of the 2011 QMJHL Entry Draft.
2011-12: Veilleux played 38 games for Victoriaville in his first season in major junior hockey and represented Canada Quebec in the 2012 U17 World Hockey Challenge. He scored 1 goal with 4 assists and was -2 with 71 penalty minutes for the Tigres. Victoriaville was second in the East Division but lost in the first round in the playoffs. Veilleux appeared in one game in the first round series with Baie-Comeau and had no points with 2 penalty minutes. He was invited to Hockey Canada's U18 summer camp.
2012-13: Veilleux provided energy and secondary scoring in his second season with the Tigres. He scored 11 goals with 17 assists in 66 games; finishing -6 and leading Victoriaville with 129 penalty minutes. Victoriaville finished fourth in the East Division and reached the second round in the playoffs. Veilleux scored 1 goal with 2 assists in eight playoff games and was -1 with 29 penalty minutes. His brother Steven appeared in 10 games in net for the Tigres. Ranked 171st amongst North American skaters, Tommy Veilleux was selected by Nashville in the sixth round (171st overall) of the 2013 NHL Draft.
Veilleux is an edgy and physical player who loves to agitate and generally annoy and pester opponents. He has a limited amount of offensive skills but he is not necessarily lacking in them either. He will have to improve his skating and strength if he wants to become a better all-around hitter and player.
With success coming to players of a similar mold like Jordan Tootoo or Andrew Shaw, Veilleux may yet have a place in the NHL as a bottom line agitator. It would go a long way if he could develop an offensive skill set or a better sense of discipline so he does not put his team behind the eight ball. He has a few years in the QMJHL with Victoriaville to improve in the areas of need.