That depends on your financial situation and your short-term goals. Your question gets a very different answer if you're trying to save cash right now than if you're trying to purchase a home in the next few years.
Goal: Long-term credit score boost
With a line of credit and a few credit cards, you can maintain a decent credit score right now if you keep making on-time payments on every one of your accounts each month. Your goal is to balance out your credit utilization, the percentage of your credit line on each card you actually use. Credit scoring algorithms pay close attention to this number, since many banks believe you're a bigger risk if you let any single account drift too close to the max.
In this case, shop for a strong balance transfer offer, but try to consolidate only about 30 percent of your new card's available credit limit. That will split the difference between the cash you'll spend on credit card finance charges and the money you will probably have to spend on increased home loan rates and insurance premiums if your credit score drops too quickly.
Goal: Short-term cash savings
On the other hand, if you're trying to preserve as much cash as possible right now, go ahead and shop for the balance transfer offer with the longest introductory period and the lowest "go-to" rate. Consolidate as much of your outstanding debt to the zero-APR card. Then pay down as much as you possibly can on any balances that remain on your leftover cards.
Here's the trick to make this "debt snowball" work. If you pay off an account, start applying the same amount you used to pay toward one of your other accounts. You'll keep your household budget the same, but you'll clear your debt much more quickly. Even though you might take a short term hit on your credit score, you'll significantly reduce what you're paying in finance charges while you work this system.