Ajmal has undergone remedial work to fix his bowling action after it was ruled illegal in September and he was suspended, forcing him out of last month's World Cup.
The 37-year-old had been key to Pakistan's success in Tests, one-day internationals and Twenty20 matches before his action was reported as suspect during Pakistan's tour of Sri Lanka in August last year.
Pakistan's new selection committee, headed by former batsman Haroon Rasheed, warned the squads that discipline would be closely watched.
“We have told the players that Pakistan comes first,” said Rasheed.
“There will be no compromise on discipline and that's why we have dropped Umar from all formats and Ahmed Shehzad from two.”
Head coach Waqar Younis had complained about Shehzad and Umar's lack of discipline during Pakistan's World Cup campaign, which ended in a quarter-final defeat to hosts and eventual champions Australia.
Umar managed just 164 runs in seven matches while Shehzad scored 222 in as many games.
Spinning allrounder Mohammad Hafeez also returns to all three formats after missing the World Cup with a calf injury.
He too has suffered problems with his action, which was ruled illegal in December, and faces a re-test on April 9 to try to get clearance to play.
The one-day squad will be captained by Azhar Ali, replacing the now retired Misbah-ul-Haq.
Misbah continues to lead the Test side while Shahid Afridi, who also retired from one-day cricket after the World Cup, will captain the Twenty20 side.
Pakistan will play two Tests, three one-day internationals and a Twenty20 on the Bangladesh tour, with the first one-dayer starting on April 17.
Fit-again pacemen Junaid Khan returns to the Test and Twenty20 squads while Umar Gul manages a Twenty20 berth.
Also dropped from the World Cup squad were Younis Khan and opener Nasir Jamshed, who both had disappointing tournaments, and paceman Mohammad Irfan, who is injured.
Promising youngsters Sami Aslam, Babar Azam, Mukhtar Ahmed and Mohammad Rizwan are included in various sqauds for the first time.
Test: Misbah-ul-Haq (captain), Babar Azam, Mohammad Hafeez, Sami Aslam, Asad Shafiq, Azhar Ali, Younis Khan, Haris Sohail, Saeed Ajmal, Yasir Shah, Zulfiqar Babar, Sarfraz Ahmed, Wahab Riaz, Junaid Khan, Sohail Khan, Rahat Ali
One-day international: Azhar Ali (captain), Sami Aslam, Sarfraz Ahmed, Mohammad Hafeez, Asad Shafiq, Fawad Alam, Mohammad Rizwan, Haris Sohail, Sohaib Maqsood, Saeed Ajmal, Yasir Shah, Wahab Riaz, Rahat Ali, Ehsan Adil, Sohail Khan
Twenty20: Shahid Afridi (captain), Ahmed Shehzad, Sarfraz Ahmed, Mohammad Hafeez, Mukhtar Ahmed, Sohaib Maqsood, Haris Sohail, Mohammad Rizwan, Saeed Ajmal, Saad Nasim, Sohail Tanveer, Wahab Riaz, Sohail Khan, Umar Gul, Junaid Khan – AFP
Original source: Ajmal back in Pakistan fold