Housing starts leaped last month, and consumers boosted their spending substantially, according to separate reports released yesterday by the Commerce Department.

Housing starts were up 30 percent last month on the May figure at a seasonally adjusted annual rate of 1.19 million houses. Personal spending rose at an annual rate of $16.7 billion in June after falling in the two previous months.

At the same time, the downward trend in interest rates which has helped to revive some sectors of the economy, especially housing, continued. Chase Manhattan Bank dropped its prime lending rate to 11 1/4 percent from 11 1/2 percent. Citibank, the nation's second-largest bank, was the first major bank to move to 11 1/4 percent last Friday. Other banks are now expected to follow Chase and Citibank.

Although the housing and spending figures suggested that the seeds of economic recovery have been sown, other government figures published yesterday showed that the recession still is biting.

Personal incomes rose by $8 billion in June to a seasonally adjusted annual rate of $2.086 trillion. This probably was not enough to offset the effect on real incomes of inflation during the month, although it marked a bigger increase than in May.

Factories operated at only 76.1 percent of full capacity in June, down 2.3 percentage points from May, the Federal Reserve Board reported yesterday.

The economic slowdown which has thrown many people out of work has forced manufacturers to cut back on their output and underutilize capacity. The June operating rate was the lowest since October 1975.

Manufacturing incomes dropped by $3.7 billion, before taking account of inflation. Increased layoffs and short-time working outweighed the effect on total earnings of an increase in average hourly pay.

But many economists and government officials believe the economy is nearing a turn after a dramatic fall in gross national product in the March-to-June quarter.

Secretary of Commerce Philip Klutznick, commented that the housing figures were encouraging "not only to builders and to potential home buyers but also to those looking for signs of economy recovery." Commerce officials called the increase in consumer spending in June "substantial."

The rebound in housing still left starts far below their level a year ago, Commerce officials reported. The housing industry has been one of the hardest hit by recession, along with the automobile industry.

Michael Sumichrast, chief economist for the National Association of Home Builders, commented that house building had fallen so far that "it could not go down more."

Nevertheless, he was surprised at the size of the June increase in starts and noted that sales, which rose in May, continued to improve in June and July. "There's no question we've reached the bottom of the housing recession," he said yesterday.

There was also a rise in the number of new building permits issued in June.

Commerce Department figures showed that these went from 825,000 at a seasonally adjusted rate in May to 1.059 million last month.

Housing and Urban Development Secretary Moon Landrieu yesterday hailed the June housing figures as an important sign that the Carter administration's economic policies are working. Klutznick called the housing industry a "bellwether," suggesting that the turnaround there could foreshadow a wider upturn.

The all in the mortgage rate was a principal reason for the pickup in home buying and building. Sumichrast said that when mortgage rates dropped below 12 percent people started to buy again.

The upturn in consumer spending last month came despite continued slack in incomes. Consumers appear to have dipped into their savings in June to finance more spending. The proportion of personal income which was saved last month fell to 4.45 percent from 5.08 percent in May.

A dramatic fall in consumer spending and a rise in savings were major elements of the economy's slowdown in the second quarter. Yesterday's figures suggest that consumer confidence may be rebuilding.

But a Commerce Department official remarked that the rise in spending last month could prove a "flash in the pan." There was not a matching rise in incomes last month to sustain it, the official said, nor was there one in sight.

The 1 percent rise in outlays was almost certainly more than large enough to offset inflation in June, which would make it the first rise in real spending since January. But the 0.3 percent increase in after-tax personal income in the month probably will translate into a small decline in real incomes after inflation.