How does paying off my mortgage affect my credit score?

Closed accounts that are charged-off remain on your credit report for 7 years from the date of first delinquency on the bill that led to the charge-off. All other accounts remain on your credit report for 10 years from the date the account is closed. Any missed payments associated with those accounts will fall off your credit report after 7 years, however.... read full answer

Accounts sent to collections (such as defaulting on a medical bill, phone bill, or loan) will remain on your credit file for 7 years from the original missed due date.

Publically available information that reflects irresponsibility (such as civil judgments and tax liens) generally remains on your credit reports for 7 years.

Bankruptcy information (Chapter 7 & Chapter 11) can remain on your credit reports for 10 years from the date filed. A discharged Chapter 13 bankruptcy generally remains on your credit file for 7 years from the date filed, while a non-discharged chapter 13 bankruptcy remains for 10 years.

If you believe that the negative information isn't supposed to be on your credit report, then you can submit a dispute with the credit bureaus either online or by mail. You'll have to explain the error in detail, and even provide evidence where possible.

There are generally two schools of thought: (1) Pay the highest interest rate balances off first so that you pay the least amount of interest during the payoff process, or (2) pay off the smallest balances first regardless of interest rate so that you see the "fruits of your labor" and don't get discouraged.... read full answer
I tend to suggest option #2. (A) Write down all of your debts, balances, and minimum payments in order from smallest to largest balance. (B) Figure out what you budget allows you to put toward the debts (at least the minimum and, if possible, more than that to get them paid off as quickly as possible). Ideally, you can limit your variable expenses (e.g. eating out, entertainment, etc.) to put as much as possible toward the debts. (C) Pay the minimum payments on all but the smallest balance debt and pay the rest that you have budgeted for debt payments to the smallest balance. (D) When the first balance is paid off, keep making minimum payments on all but the smallest remaining debt, and move the money that you were putting toward the first one to the second one. (E) Repeat the process until the debts are paid.
You may pay more in interest using this method, but it will help avoid discouragement if you work on higher balances first and don't chip away as quickly. And at the end of the day, if you get discouraged and quit the process, then you will not end up clearing the debts.
Hope this helps and good luck.