Filing Married Filing Separately generally provides a disadvantage on your federal income taxes, due to the elimination of the "marriage penalty" (extra tax burden from filing as a married couple). If you are both working, then it is possible that filing married filing separately would work out to be advantageous on your federal income tax return, but you would have to be making very similar amounts of money and above $70k each, and not have any of the deductions and credits that are eliminated when you elect to file separately (generally, education credits and student loan interest, dependent care credit, earned income credit, etc.).
Either spouse my claim the dependents on their tax return, but it is generally a better idea for the higher income spouse to claim the dependents, as long as their separate income is under $200,000. Other considerations might be looking at who owes without the children, and electing to put the children there, in case any refund you receive from the other married filing separate tax return is delayed. In other words, don't be put in a situation where you are waiting for a big tax refund and have to pay the IRS a big tax bill, in case the big tax refund is delayed.
A couple of things to add: Married Filing Separate DOES create an advantage for many states. States like Ohio have a large marriage tax penalty, and as long as both spouses are working, there is almost always an advantage to file married filing separately. You have to look at both to determine if there is a net benefit - it might cost an extra $200 to file separately at the federal level, while increasing the state tax return by $800, for a net $600 advantage to filing separately.
In addition, once you have filed jointly, you cannot amend your taxes and file separately. You can, however, amend your married filing separately returns to a married filing jointly return.