Paulina Rubio to Estranged Husband -- You're a Bad Dad ... My Private Eye Proved It!

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.