Wales full-back Davies, 21, joins on a five-year deal and Netherlands goalkeeper Vorm, 30, has agreed a four-year contract with the Londoners.
Iceland international midfielder Sigurdsson, 24, has signed a four-year deal with Swansea.
He spent 2011-12 on loan at the Liberty Stadium from German club Hoffenheim, scoring seven goals in 18 appearances.
Spurs are thought to have fought off stiff competition for Davies, with Liverpool boss Brendan Rodgers understood to be an admirer.
But Davies said Spurs were "the ones chasing me from the beginning" and that joining the north London club was "too good an opportunity to turn down".
He told Swansea's official website: "I had never thought or wanted to leave Swansea, but an opportunity like this doesn't come up often in your career."
Davies made his debut for the Swans as a 19-year-old in August 2012 and quickly established himself as first-choice left-back when Neil Taylor suffered a long-term injury.
His performances impressed Wales manager Chris Coleman, who handed the Neath-born player the first of his 10 caps in the 2-1 win over Scotland in October 2012.
Vorm was part of the Dutch squad that finished third at the Fifa World Cup in Brazil earlier this month, but his future has looked in doubt at Swansea ever since former Arsenal keeper Lukasz Fabianski was signed in May.
He will compete with France number one Hugo Lloris and 43-year-old Brad Friedel for selection at Spurs.
|Gylfi Sigurdsson Factfile|
|Born: 9 September 1989, Reykjavik|
|Clubs: Reading, Shrewsbury Town (loan), Crewe Alexandra (loan), 1899 Hoffenheim, Swansea City (loan), Tottenham Hotspur|
Sigurdsson arrives at the Liberty after spending two seasons at White Hart Lane, scoring eight goals in 58 top-flight appearances for the London club.
He said: "It was tough leaving a club that I've enjoyed playing for over the last few years, but as soon as Swansea came in it was an easy decision to make."
Spanish forward Michu has left Swansea to join Napoli on loan, while winger Pablo Hernandez is believed to be on the brink of a move to Al Arabi in Qatar.
The Welsh club are also thought to be close to completing a deal to sign Ecuador World Cup winger Jefferson Montero from Mexican club Monarcas Morelia.