Terms on mortgages vary. There exist both "recourse" and "non-recourse" loans. This applies to loans other than mortgages, but it seems rare to see anything but a mortgage of the "non-recourse" type. Note that these types are general descriptions, and these terms are not always used in the loan documentation. The loan documentation will spell out specific terms that people then lump into one of these categories based upon the practical implications of the terms.
Older first mortgages were often of the "non-recourse" type. In this case, the bank's sole remedy for you failing to pay is that they can either hassle you until you do or take the collateral (the property backing the loan). The "non-recourse" part says that once they've exhausted that option of taking the property, they have no further recourse. They get the property, but they're done.
"Recourse" loans allow further recourse beyond seizing the collateral that backed the loan. 2nd (and 3rd, etc.) mortgages, auto/boat/aircraft loans, and personal loans are generally of this type. These types of terms allow the lender to take further action above and beyond that specified in the loan to satisfy the debt. These actions could include seizing additional property, garnishment of wages, etc. A court would have to approve such actions, but it's not too hard to get a wage garnishment order if you can demonstrate an outstanding debt whose terms permit such actions. Lots of newer 1st mortgages these days, especially "work out" mortgages that are issued when people call behind on an existing one, are also of the "recourse" type. Most "subprime" and "non-traditional" (e.g. ARM, interest only, balloon payment etc.) 1st mortgages issued in the past were also of this type.
I don't have the figure to say which are more common. I'd suspect that most of the defaults and resulting foreclosures these days are from newer loans as well as non-traditional loans, so they're likely to permit recourse beyond seizing the collateral. I think that some states also place regulations that are inconsistent with other states upon mortgages. For example, in some states, 1st mortgages with recourse terms may be prohibited while, in other states, non-recourse terms may not be allowed.
On a semi-related subject, if you ever settle a debt for less than the amount owed, make sure you pay it by some means where you have a duplicate of the payment (e.g. carbon copy of a cashier's check), and write the words "PAID IN FULL WITHOUT FURTHER RECOURSE" on the payment. Once that payment is accepted, the lender can no longer come after you. If you just have some side agreement, they can easily accept your payment then deny the agreement exists (leaving you to attempt to prove it does - good luck) leaving you on the hook for the rest of the debt. I also recommend using USPS registered mail with restricted delivery and return receipt. That makes it REALLY hard for them to deny they got the payment, and it can also bring "mail fraud" up if they do something really dumb like cash the check but not credit you. It'll cost you like $20 to mail it, but it's worth it for the protection. You can save a few bucks with certified and get almost the same protection against denial of receipt, but registered offers greater protection against loss of the check.