Rich, I volunteer for the largest cairn terrier rescue in the world, I foster and I am an Intake Coordinator. Cairn Terriers are on the list of dogs that don't shed dander. I have two cairns myself, and normally a third one resides in this house when I foster. Even with 3 cairns running around, I have little to NO dog hair on my floors, but I do groom them properly (strip them, not cut) about 4 times a year. However, unless you've owned a cairn terrier before, and use the right buzz words on the application (crate train, human alpha, etc), they will not adopt to a person with a child under the age of 9. Cairns are prey driven, and love their toys, and many times they will go after a toy and mistakenly grab a little finger. In the correct household, with strong human alpha's, this behavior can be diminished, but it takes work. I did read your considering a Bichon, but also saw a post on cairns......
The rescue band wagon is great, we all need to rescue a dog, however, most don't realize what that entails. Unless you can find a rescue that has whelped puppies w/in the rescue, and this is a rarity, odds are your going to get a 1-2 year old dog. Many rescue dogs come with baggage. Besides being "poorly bred" which often means health problems, there are behavior problems. If you find a good rescue org, they will have foster homes that have the knowledge and are equipped to handle these problems, and won't let them go to a "forever" home until these problems are addressed and well on their way to being solved. Just make sure you ask the right questions. If you want, pm me, I can help you with those questions.
Now, the disclaimer, I am in no way saying do not go to a rescue. I am saying you need to do your homework before adopting, not only on the breed, but also on the rescue and the individual dog your considering adopting.