Child custody situations are sometimes complex because of the acrimony between the parents. The children can pick up on this, so they may try to capitalize on the situation. It is imperative that all parents take the time to learn the signs that their children are manipulating them.
Some children learn early that what one parent disapproves of, the other will approve of. Once they realize this, they will try to figure out which person will give them their way. This is human nature, but that doesn't mean that you and your ex need to allow it to happen.
Tailoring requests to the parent
Children who tailor their request to the parent they are asking might be playing one against the other. The child might try to get a quick answer because they don't want the parent to have a chance to contact the ex. This is a bad sign that should result in your backing up a bit to find out the truth of what is going on.
Multiple requests for the same thing
A child might ask one parent for something. When they don't get what they want, they go to the other parent. This is where communication comes into the picture. If you and your ex are letting each other know what's going on, especially with big decisions, there is less of a chance that the kids will be able to successfully do this.
Open the lines of communication
The lines of communication should always be open between you, your ex and the kids. This lets everyone talk about things that are happening so that everyone is on the same page. Remember to keep things civil between the adults so that you are modeling a good relationship.
Present a united front
You aren't always going to agree with the decisions your ex makes and they might not agree with yours. Unless there is a serious safety issue, you should both back each other up in front of the children. You may discuss the differences when they aren't around so that you still appear to be a parenting team in front of them. Not only does this shore up the fact that they can't play the adults against each other, it also solidifies the stability of the situation.
For big issues that come up, having a plan put into the child custody paperwork might be a good idea. The parenting plan should also contain methods of resolution for times when there is a disagreement.