In Massachusetts, as soon as a married couple files for divorce or custody, the couple is deemed to have temporary shared legal custody of any children.  The Probate and Family Court can alter that arrangement and grant custody to one parent if the court finds that the joint custody is not in the best interest of the child. 

As to your second questions, the answer, one that seems vague unsatisfying, is that the Probate and Family Court will seek a custody arrangement that is "in the best interest of the child."  Below, I have reproduced some of the more pertinent portions of Massachusetts General Laws Chapter 208, Section 31, including the definitions of the various forms of custody:

“Sole legal custody”, one parent shall have the right and responsibility to make major decisions regarding the child’s welfare including matters of education, medical care and emotional, moral and religious development.

“Shared legal custody”, continued mutual responsibility and involvement by both parents in major decisions regarding the child’s welfare including matters of education, medical care and emotional, moral and religious development.

“Sole physical custody”, a child shall reside with and be under the supervision of one parent, subject to reasonable visitation by the other parent, unless the court determines that such visitation would not be in the best interest of the child.

“Shared physical custody”, a child shall have periods of residing with and being under the supervision of each parent; provided, however, that physical custody shall be shared by the parents in such a way as to assure a child frequent and continued contact with both parents.

In making an order or judgment relative to the custody of children, the rights of the parents shall, in the absence of misconduct, be held to be equal, and the happiness and welfare of the children shall determine their custody. When considering the happiness and welfare of the child, the court shall consider whether or not the child’s present or past living conditions adversely affect his physical, mental, moral or emotional health.

In determining whether temporary shared legal custody would not be in the best interest of the child, the court shall consider all relevant facts including, but not limited to, whether any member of the family abuses alcohol or other drugs or has deserted the child and whether the parties have a history of being able and willing to cooperate in matters concerning the child.

There shall be no presumption either in favor of or against shared legal or physical custody at the time of the trial on the merits . . . 

In addition to the factors specifically mentioned above, judges will consider all factors that are relevant to the "best interest" determination.  So, for example, if one parent has a history of abuse, that fact would obviously weight heavily in favor of granting custody to the other parent.  Attorneys who work in this field tell me that, like most of us would, judges tend to take a common sense approach to this decision and will also consider other factors, such as:(1) who has acted as the primary caregiver and how the child has fared under his or her care; (2) how the decision would effect the schooling of the child and whether it might force a change in schools; and (3) which parent is available to spend more time with the child.  With older children, the court might also consider the wishes of the child, though not as a deciding factor.

Rather than leaving this important decision up to the courts, parents can enter agreements regarding the custody of their children and, as long as the court does finds that arrangement proposed in the agreement is in the best interest of the children, it will enter an order in accordance with the agreement.  For more information of to post a question, visit our MA Divorce Discussion Forum.