Hi K.. First of all, Thank you for not believing in buying a puppy! Like Erika, I am also involved in Rescue and have seen many families turned down for just the reasons she mentioned. There are some things that will automatically cause you to be rejected and many others that are "red flags" that you can get around with a good answer/explanation. Please don't take offense at my answers below, but as I don't know you, I'll just list some common things people do or say that can hurt you on applications. Maybe none of these apply to you, but maybe they'll help someone else.
The #1 thing I have seen people turned down for is the "vet check." If you have had animals now or in the past, the rescue group will require you list the vet you used on the application. They will call that vet to verify that you had the pet spayed/neutered, kept shots up to date and kept dogs on heart worm preventative as well as followed doctors orders for routine care and medications. We have turned people down because people feed dogs people food instead of dog food and/or allowed a dog to be obese. (A little extra weight won't hurt, but if it concerns a vet, it will concern us!)
Having a young child may very well hurt your chances with many dogs. It depends on which group you are applying to as to how stringent rules are. Several have a "no kids under school age" rule no matter what, which may have excluded you as a potential adopter immediately. Others go by the particular dog and their knowledge of the animal. It may well depend on which dog you are "applying" for. I have also seen people who put on the form that a young child will be responsible for daily care or that the pet is specifically for the child. I do agree that it can be a great way to teach a child responsibility, but you need to be the one ultimately responsible and say that on the form.
If you don't own your home, you'll need proof your landlord has approved pets. It is also very hard to be approved if you don't have a fenced-in yard. There are a few exceptions for small dogs, but again it is a huge red flag.
Do you both work full time? Sometimes this can also hurt your chances. Most groups have a set number of hours that they expect you to be home. Again, it is usually dependent on the particular dog you want, but if you say you are out of the home 12 hours a day, you'll pretty much be rejected off the bat. People get around this sometimes by agreeing to a dog sitter or doggie day care a few days a week or even just going home for lunches, etc. It is simply something to discuss with the adoption counselor.
Usually applications ask if you are willing to house train and/or obedience train a new dog. Even if you plan on getting one that is already trained, answering "No" to these questions reflects poorly, so always say YES! And keep in mind many rescue dogs are big questions marks in these areas. I have adopted one that they said was not trained, but he has not once messed in the house in 10 years, and another they said was trained who turned out to be only partially trained.
Also, be aware that many rescue groups are 100% volunteer based, so the "service" you receive may not be stellar. You could have encountered an overzealous volunteer that turns down anyone who is not perfect. Obviously perfection is hard to come by. Do you have your heart set on a particular breed? This could hurt you also. For example, you'd be hard pressed to get me to approve a Chihuahua for a home with young kids. (And I have an almost 2 year old and 2 Chihuahuas!) If you just want a good family dog and you have everything above in order, I would suggest contacting a rescue group and telling them exactly how you feel. A good group will want to work with you in finding the right match. Try http://www.fmhs.org/adopt/dogs.asp They are strict (as any reputable group should be), but also fair and try very hard to match dogs to good homes. With them, I highly suggest a phone call to discuss your situation before or at the time you apply. They do adoptions at the PetSmart in Flower Mound. Go in person to talk to them. Even if they don't currently have a dog you are interested in, they may soon. New dogs come in every week. Feel free to email me any specific concerns or questions you may have.