The 33-year-old international was a free agent after his contract expired at Goodison Park in the summer of 2022.
There were reports that MLS outfit DC United, who are managed by Wayne Rooney, were interested in the midfielder but talks fell through and he will now continue his career in Denmark.
Lyngby ply their trade in the Danish Superliga where they are currently ninth-placed in the league standings. They are managed by Icelandic coach Freyr Alexandersson who is well known to Sigurdsson as he has previously worked as Iceland's assistant manager.
"I hope that I can come in and contribute with a lot of experience and quality," Sigurdsson told the club's official media.
"I know that there are a lot of young and talented players in the team, and I really want (them) to learn from me so that we can lift ourselves up together.
"I have heard about the good support we have, and I am very much looking forward to meeting the fans and not least playing in front of them," he added.
Sigurdsson's final professional appearance was during Everton's 5-0 loss to Manchester City in May 2021. He has made 318 Premier League appearances, scoring 67 goals and 52 assists.