WASHINGTON, May 21 (Reuters) - U.S. single-family homebuilding dropped sharply in April and permits for future construction fell, suggesting the housing market could remain subdued for a while as the Iran war drives up mortgage rates and an oversupply of new houses persists. 

Single-family housing starts, which account for the bulk of homebuilding, tumbled 9.0% to a seasonally adjusted annual rate of 930,000 units, the Commerce Department's Census Bureau said on Thursday. Single-family homebuilding fell in all four regions. It declined 2.4% year-on-year in April.

The U.S.-Israel war with Iran has raised oil prices and is fanning inflation, driving up U.S. Treasury yields. Mortgage rates track the 10-year Treasury note, whose yield is hovering near a 1-1/2 year high. 

The popular 30-year fixed mortgage rate averaged 6.36% last week, data from mortgage finance agency Freddie Mac showed. It averaged 5.98% at the end of February, when the war started, as Freddie Mac and Fannie Mae expanded purchases of mortgage-backed securities. 

Permits for future construction of single-family homes dropped 2.6% last month to a rate of 872,000 units. They decreased 5.5% year-on-year in April.

Homebuilding is also being weighed down by tariffs on imported goods, including lumber and vanity cabinets, as well as higher land, labor and construction costs. Residential investment, which includes home building, has contracted for five straight quarters.

A National Association of Home Builders survey this week showed homebuilder sentiment remaining depressed in May.

Though new housing inventory has declined from levels last seen in late 2007, it remains elevated.

Starts for housing projects with 5 units or more, a very volatile segment, jumped 14.3% to a rate of 529,000 units in April. Multi-family housing starts shot up 23.3% year-on-year. Overall housing starts fell 2.8% to a pace of 1.465 million units. They increased 4.6% year-on-year in April.

Building permits for multi-family housing projects surged 22.7% to a rate of 514,000 units in April. Overall building permits increased 5.8% to a rate of 1.442 million units last month. They fell 0.2% year-on-year in April.