1. Do not let them in, no matter how much they ask. Animal Control generally cannot enter your home without a warrant, or your permission. While regular police can enter in emergency situations when human life is at risk (i.e. they hear gunshots and a scream inside), there are few, if any, situations in which Animal Control can enter your home without a warrant. Simply tell them they may not come in.
2. If you let them in, anything they find in "plain sight" can be used against you. In some circumstances Animal Control officers, unable to find a legitimate reason to make an arrest, have reported building or zoning violations. This may include caging you attached to a wall without a building permit, that extra outlet in the puppy room, having more pets than allowed by zoning, even extension cords in violation of fire codes! No matter how clean your kennel, if they want to find a violation, they will.
3. Do not talk to them from an open doorway. Step outside and close (and lock if possible) the door behind you. This is necessary because:
A) Anything they see through the open door is "plain sight" and may be the basis for an arrest, or probable cause for a search warrant.
B) If they make an arrest or even feel threatened they are usually permitted to search for weapons in your immediate area. Do you keep a baseball bat inside the door for your protection? Even if you don't, once they step inside to look, they are in your home and may continue to search.
C) It is hard not to be intimidated by someone in authority. Some animal control is even done by local police, who carry guns. It is easy for them to get "in your face", causing you to back up into the home. Once you go in, it will be interpreted as an invitation to follow.
4. If they claim to have a warrant, demand to see it. In general, a search warrant must be signed by a judge. A warrant to search your home for dogs does not include an inventory of your jewelry box. A warrant to search your kennel in the garage or in the barn does not include a search of your home.
5. In some locations dog owners may have obtained special "breeder permits" that stipulate that Animal Control has your permission to enter at any time. If you have signed such a permit they still cannot enter against your wishes, since you can revoke the permission at any time. However, if you refuse permission it may allow them to cancel your breeder permit, so you have to weigh the consequences.
6. Warning - anyone in lawful possession of the premises may be able to give permission for a search. Make sure your roommate, babysitter, dog-sitter, housekeeper and others know that they should not let animal control into your home or on your property (i.e. backyard, garage, etc.).