Brighton have lifted Mauricio Taricco's suspension but are continuing their inquiry into Gus Poyet and Charlie Oatway.
The Championship club suspended their management trio in the aftermath of the 2-0 play-off semi-final defeat by Crystal Palace last month.
Brighton now say they are looking forward to welcoming Taricco back to work but the future of Poyet, who has been linked with the vacant job at Wigan, remains unclear.
A club statement read: "In relation to the suspensions of manager Gus Poyet, assistant manager Mauricio Taricco, and first-team coach Charlie Oatway, the club can confirm the following.
"The internal process concerning Mauricio Taricco has now concluded. Mauricio's suspension has been lifted. The club looks forward to Mauricio's return to work following his scheduled annual leave.
"The club's internal processes concerning Gus Poyet and Charlie Oatway are on-going. As such, the club is not able to make any further comment on this matter at this time."
The former Rangers and Everton defender, 43, had been working as a coach at Goodison Park after retiring as a player in 2012.
Weir replaces Danny Wilson, sacked in April with five games of the regular season remaining, and becomes the Blades' seventh permanent manager in the last seven years.
Weir told the club's website: "My first thoughts about the job are of excitement, I understand how privileged I am to be named manager of a fantastic football club like Sheffield United.
"I realise what a great opportunity is in front of me now and my message to the supporters is that I will work hard and give everything to be successful at Bramall Lane.
"I honestly can't wait to get started and get a real feel for the club, then obviously I am eager to get organised for when the players return for pre-season training later this month."
Blades coach Chris Morgan had been placed in temporary charge following Wilson's departure and the former United defender had been hoping to land the job.
Central Coast Mariners boss Graham Arnold had also been interviewed for the post along with former Blades midfielder and coach Stuart McCall, who opted to remain as Motherwell manager.
Another former player at Bramall Lane, Port Vale coach Robert Page, ex-Blackburn, Blackpool and Portsmouth manager Michael Appleton and former Middlesbrough boss Gareth Southgate were also reported to have been on the club's shortlist.
Chief executive Julian Winter said: "We are thrilled to appoint David and the three-year contract emphasises the fact that he will assist in a change in club culture.
"I have stated before that the club needs to act differently and David is young, has a great pedigree and will bring a new vision and leadership to the club.
"The process of appointing a new manager has taken slightly longer than we envisaged but by taking our time we believe we have the attracted the right man to take Sheffield United forward."
Weir, who began his playing career at Falkirk before a three-year spell at Hearts, moved south of the border to join Everton in 1999 and went on to make over 230 appearances for the Merseysiders.
He returned to Scotland in 2007 with Rangers, where he made over 140 appearances in five years before announcing he wanted to extend his playing career back in England.
Weir trained briefly with the Blades, but instead returned to Everton in early 2012 to join the club's coaching staff.
He is understood to have been interviewed for the post of manager at Goodison Park following David Moyes' departure to Manchester United.
Weir won 69 caps for Scotland and became the oldest ever international player for his country when he appeared against Lithuania in September 2010, aged 40 years and 116 days.
