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.
2013-14: Veilleux did not make his 2013-14 debut with Victoriaville until mid-November after suffering a broken collar bone during a rookie game with Nashville prior to the season and also missed January and most of February due to injury. Skating in 17 games for the Tigres in his third QMJHL season the 18-year-old scored 4 goals with 4 assists and was +4 with 31 penalty minutes. Veilleux played in two games during the Tigres five-game playoff series with Drummondville; finishing an even plus/minus with no points and 4 penalty minutes.
Veilleux is an edgy, physical player who can agitate and pester opponents. He showed some offensive elements to his game at times but his calling card to this point has been his physical play and a willingness to drop the gloves if necessary. Injuries hampered his development in two of his three junior seasons and he may need to be less reckless to have success — and remain healthy — at the pro level.
Nashville selected Veilleux late in the 2013 NHL Draft based on his hitting ability and a style that fit in well with the "Smashville" mentality long promoted in the past. With several highly-skilled skaters now either on the cusp of joining the Predators or in the team's prospect pipeline, Veilleux projects as the type of player who can be an enforcer to protect some of those players and someone who can crash into the tight areas to score goals in high traffic areas. His injury history is a concern and may prevent him from reaching his long-term potential.