Two SABR members, W. Madison McEntire and John Rickert, put together a nice spreadsheet of all teams in the majors that had winning or losing streaks of at least 10 games from 1900 through 2004.
Some of the highlights pointed out to me by Madison were:
The Braves never had a 10-game winning steak while in Boston note, from 1900 on
The 1999 San Diego Padres have the longest winning streak by a sub-.500 team (14 games!).
The 1951 New York Giants are the only team to lose 10+ in a row and even PLAY in the World Series.
The 1906 Chicago Cubs are the only team to have four 10+ game winning streaks in one year.
Other things I noticed from the list.
You would expect a team that wins the World Series to have had a 10-game winning streak. But how many have had more than one?
The 1909 Pittsburgh Pirates, 16 and 14 games, the Pirates were 110-42 that season.
The 1939 New York Yankees, 12 and 10 games, the Yankees were 106-45.
The 1955 Brooklyn Dodgers, 10 and 11 games, the Dodgers were 98-55.
The 1969 New York Mets, 11 and 10 games, the Mets were 100-62 and had never won 10 straight before.
The Dodgers, current owners of a 10-game winnning streak, have their first one since May 14-24, 2003. During that stretch, the Dodgers never scored more than 6 runs nor gave up more than 4. They gave up one run seven times during the streak.