The U.S. magazine that almost caused Courtney Love and Kurt Cobain to lose custody of their child was 'Vanity Fair'. Love and Cobain came very close to losing custody of their daughter Frances Bean Cobain because of their personal conduct around the child and before she was born.

In 1992, when Frances was just a few weeks old, child welfare officers removed her from her parents’ home for a short period of time following the publication of a Vanity Fair article that alleged Love had used drugs (heroin) during her pregnancy and after her daughter's birth. The investigation was eventually dismissed and Frances was returned to Love and Cobain.

At the time it was reported in local newspapers that law officials, a judge, Kurt Cobain's mother, Kurt's sister. and Frances Bean Cobain also thought Courtney Love was a "bad parent".

More Info: en.wikipedia.org