Some of these sources are free, some are pay sites and some are pay sites which offer a free trial or free weekend. My strategy for using the latter is to keep track of all the research I want to do and then spend a busy weekend looking everything up at once.
General sources
FindMyPast - Pay site beyond a basic search
Newpapers.com - often  has free weekends and has regular free trials
Google Search - Newspaper list from A to Z
Genealogy Bank - Newspapers from 1690 to the present
Newspaper Archives - Free trials and free weekends
Small Town Newspapers - Search over 300 small-town newspapers, some dating back to the 19th century, for free.Â
Advanced Archives - Â specializes in converting newspaper microfilm to a searchable digital archive Over 100 million newspaper pages from all over the United States, plus a few from Canada digitized so far.Â
Elephind - Newspaper searchÂ
Fulton History - over 55 million digitized newspaper pages
Google Newspaper Archives - a huge selection of digital newspaper archives from around the world
CanadaÂ
 Family Search's index of Canadian Newspapers plus much more
UBC's collection of small town BC Newspapers - an extensive collection of early newspapers, added to regularly
The Peel Prairie Provinces collection of newspapers - an excellent sourceÂ
Online access to major BC papers through ProQuest BC Historical Newspapers  - The Vancouver Sun, Province and Victoria Times Colonist archives
Simon Fraser University Digitized Newspaper Archives
Ontario Community Newspaper Portal
INK - Our Digital World (ODW) Newspaper Collection
Ontario Newspaper Achives - Bowling Green University - links to Ontario newspapers
Google News Archives Canada - a wide selection of Canadian newspaper archives
Britain
USA
Chronicling America - Library of Congress Newspaper Archives
Family Search USA newspaper links
USA Newspaper listings by state
California: California Digital Newspaper Collection, The City of Monterrey: Historical Newspapers
Colorado: Colorado Historic Newspapers Collection
Connecticut: Connecticut Digital Archive
Delaware: New Castle Public Library
Florida: Florida Digital Newspaper Library
Georgia: Georgia Historic Newspapers
Hawaii: Papakilo Database
Idaho: Marshall Public Library
Illinois: Illinois Digital Newspaper Collections
Indiana: Hoosier State Chronicles
Kansas: Kansas Memory
Kentucky: Kentucky Digital Newspapers and Microfilm, Kentucky Digital Newspapers
Louisiana: Louisiana Digital Library
Maine: Maine Newspaper Project
Maryland: Worcester County Library
Massachusetts: Boston College Newspapers, Cambridge Public Library, Sturgis Library
Michigan: Digital Michigan Newspapers
Minnesota: Minnesota Historical Society
Mississippi: Carnegie Public Library of Clarksdale, Newspaper Archive of The Woodville Republican
Missouri: Missouri Digital Newspaper Project
Montana: Montana Newspapers
Nebraska: Nebraska Newspapers, NebraskAccess
Nevada: Nevada Library Cooperative
New Hampshire: NH History
New Jersey: New Brunswick Free Public Library
New Mexico: New Mexico Historical Newspapers
New York: Brooklyn Newsstand, HRVH Historical Newspapers, NYS Historic Newspapers
North Carolina: North Carolina Newspapers
North Dakota: Leach Public Library, NDArchive
Ohio: Ohio Memory
Oklahoma: The Gateway to Oklahoma History
Oregon: Historic Oregon Newspapers
Pennsylvania: Alexander Hamilton Memorial Free Library, Pennsylvania Newspaper Archive
Puerto Rico and the Caribbean: Digital Library of the Caribbean: Caribbean Newspapers
Rhode Island: Pawtucket Public Library
South Carolina: Historical Newspapers of South Carolina
South Dakota: Grant County Public Library
Tennessee: Chattanooga Newspapers
Texas: The Portal to Texas History, Texas Tech University DSpace Repository
Utah: Utah Digital Newspapers
Vermont: Arvin A. Brown Public Library Community History Archive
Virginia: Virginia Chronicle
Washington: Washington Digital Newspapers
West Virginia: Potomac State College West Virginia Newspapers
Wisconsin: Irvin L. Young Memorial Library
Wyoming: Wyoming Digital Newspaper Collection
Australia
Trove - completely free access to Australian newspapers