Absolutely bank and credit card statements are acceptable as proof of payment for expenses; just as are actual receipts or invoices from the suppliers and service providers. This is true as long as your bank or credit card company shows the "payee" by name, instead of just a date and a dollar amount (charge) of the amount paid. You can think of detailed bank and credit card statements as being very much similar to copies of cancelled checks. They are a reliable, third party, verification of expenses paid.
Of course, if you have actual invoices or receipts as well, then please keep them too for your records (you can even scan them and save them to a thumb drive or other computer storage device, if you don't like keeping and storing paper documents).
That said, as a CPA, I have personally been through enough tax audits at both the federal and state level to know that bank and credit card statements will certainly be accepted as proof of payment (for expense reporting purposes) on both a Schedule C Self-Employment tax return, as well as all other types of tax returns and forms (e.g., school tuition expenses, business asset purchases, employer-related job expenses for hotels and meals, etc.).
Thank you for asking this important question.
