Mexican pop star Paulina Rubio was so determined to prove her husband's a bad father ... she hired a freaking private eye to track his every move -- TMZ has learned.
According to court docs, Rubio filed an emergency motion in Miami last month to strip estranged hubby Nicolas Vallejo-Nagera of all custody rights.
Rubio claims she hired a private investigator ... who exposed Nic as he "repeatedly demonstrated detrimental behavior and a lack of parental responsibility." This includes:
-- Taking the kid outside while he was sick (she asked him not to)-- Refusing to change his diaper all day-- Leaving him in his carriage totally unsupervised-- Smoking around him again and again
Rubio claims Nic showed a "total disregard" for their son -- but according to the docs, they eventually struck a compromise in which Nic agreed to keep a nanny around while he watched the kid.
We reached out to Nic's lawyer for comment -- so far, no word back.