New York (including Long Island and the Jersey Meadowlands) had a good run from May 1976 through January 1987, capturing 9 titles in all the major sports: Nets 1976, Yankees 1977-78, Islanders 1983-1986, Mets 1986 and Giants 1986.
Detroit had a great year in 1935 before pro basketball existed, winning the titles in NHL, MLB and NFL.
Oakland had a nice run in the mid-70’s:
1974: A’s win last of 3 titles.
1975 Golden State Warriors win NBA crown.
1976: Raiders gain redemption by slaughtering Vikes in the Super Bowl (technically in 1977).
To be picky, the Islanders’ championships were from 1980 to 1983.
That’s not being picky, it is being accurate. Thanks for the correction!
Also 1976: The Seals bolt for Cleveland. 
No city’s sports teams have won all four current major league sports trophies in one year. Los Angeles has gotten closest when when the Raiders, Lakers, and Dodgers all won in 1988. If we include predecessor leagues, in 1927 New York had victories from the Yankees, Giants and Brooklyn Celtics and in 1928 from the Yankees, Rangers and Celtics. The Brooklyn Celtics were part of the old American Basketball League.
Combining those two, years, of course, gives New York a two-year run of major league championships in football, baseball, hockey, and basketball. Since there are no one-year runs, that’s the shortest run for all four championships.
Uh, the Raiders didn’t even make the playoffs in 1988.
The only Super Bowl won by an LA-based team was in 1984.
The Washington, DC area had a nice run with the Bullets winning the NBA title in 1978 and the Redskins taking the Super Bowl in 1983. There was no DC baseball team at the time so the “local” team were the Baltimore Orioles, who won the World Series also in 1983. Alas, the Washington Capitals were not good at the time, only making the Stanley Cup finals once in 1998, losing to Detroit 4-0.
“at the time”? 
Edmonton had an interesting Stanley/Grey cup run in the 70s and 80s. The Edmonton Eskimos won the Grey Cup in 1975, then 5 consecutive times 1978-1982, then again in 1987. In the meantime, the Oilers won the Stanley Cup in 1984 and 1985, then 1987, 1988 and 1990. So from 1978 to 1990, Edmonton only had 3 years - 1983 (Islanders and Argonauts), 86(Habs and Ti-Cats) and 89 (Flames and Roughriders) - where they weren’t celebrating one or the other. Rather impressive.
Edmonton has no NBA or MLB team, though.
They did have a baseball team in the Pacific Coast League. No dice on 1983, 86 or89 tho; the Trappers won four PCL championships but not in those years.
LA Lakers - June 2002
Anaheim Angels - October 2002
Anaheim Ducks - 2007
The Atlanta Braves - 1995
…er…
The Cleveland Indians won their last World Series in 1948. The Cleveland Browns won the AAFC championship that year (as well as all four years in the AAFC (1946-9) and their first year (1950) in the NFL). The Cleveland Barons won both the regular season title and the Calder Cup in the AHL in 1947-8.