Ince has been out of work since resigning as MK Dons manager in May. But Trew had no doubt the former England international midfielder will prove a good appointment.
Trew added: "There were a number of high-calibre applications for the position and it was encouraging to see just how attractive a proposition managing Notts County remains to some very prestigious names in the world of football management.
"However, when we met Paul it quickly became clear that he was the ideal candidate, he'd done his homework, had some big ideas and was ready to get stuck in straightaway."
Short only took over in the summer following Cotterill's departure to Portsmouth but lost his job with the Magpies sitting 16th after 13 games in League One.
Ince started his managerial career with Macclesfield in October 2006, with the club winless after 15 games and seven points adrift at the bottom.
He led them to safety against the odds that season, only losing eight of his 35 games in charge.
He left for MK Dons in the summer, whom he led to promotion from League Two and the Johnstone's Paint Trophy in his sole season there, before joining Blackburn Rovers.
Ince's spell at the Premier League club was less successful and he was sacked after just six wins in six months.
He rejoined the Dons in July 2009 but left by the end of the season because of a cut in his budget.
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