Normally the courts look to the "best interest of the child" and want both parents in a child's life.
Here is the statute for Nebraska law:
Neb. Rev. Stat. Ann. § 43-292 (Michie Supp. 1998)
The court may terminate all parental rights between the parents or the mother of a juvenile born out of wedlock and such juvenile when the court finds such action to be in the best interests of the juvenile and it appears by the evidence that one or more of the following conditions exist:
The parents have abandoned the juvenile for 6 months or more immediately prior to the filing of the petition;
The parents have substantially and continuously or repeatedly neglected and refused to give the juvenile or a sibling of the juvenile necessary parental care and protection;
The parents, being financially able, have willfully neglected to provide the juvenile with the necessary subsistence, education, or other care necessary for his or her health, morals, or welfare or have neglected to pay for such subsistence, education, or other care when legal custody of the juvenile is lodged with others and such payment ordered by the court;
The parents are unfit by reason of debauchery, habitual use of intoxicating liquor or narcotic drugs, or repeated lewd and lascivious behavior, in which conduct is found by the court to be seriously detrimental to the health, morals, or well-being of the juvenile;
The parents are unable to discharge parental responsibilities because of mental illness or mental deficiency and there are reasonable grounds to believe that such condition will continue for a prolonged indeterminate period;
Following a determination that the juvenile is without proper parental support or care, and reasonable efforts to preserve and reunify the family if required under the law, under the direction of the court, have failed to correct the conditions leading to the determination;
The juvenile has been in an out-of-home placement for 15 or more months of the most recent 22 months;
The parent has inflicted upon the juvenile, by other than accidental means, serious bodily injury;
The parent of the juvenile has subjected the juvenile to aggravated circumstances, including, but not limited to, abandonment, torture, chronic abuse, or sexual abuse; or
The parent has committed murder of another child of the parent; committed voluntary manslaughter of another child of the parent; aided or abetted, attempted, conspired, or solicited to commit murder, or aided or abetted voluntary manslaughter of the juvenile or another child of the parent; or committed a felony assault that resulted in serious bodily injury to the juvenile or another minor child of the parent.