Personal income of Americans rose 0.3 percent in January, the smallest monthly increase in a year, the Commerce Department reported yesterday.

The department also said business inventories declined in December for the first time in a year, despite vigorous sales.

The increase in income, which followed rises of 1.2 percent in November and 1.5 percent in December, was small because of a variety of unusual factors, the department said.

Overall, the total of wages, salaries, dividend payments, transfer payments such as Social Security and Welfare benefits and other income rose by $4.3 billion to an annual rate of $1 trillion, 626.4 billion.

Despite the weak showing, wages and salaries rose $10.9 billion, or 1 percent, after rising only $3.8 billion in December. The report takes into account normal first-to-the-year wages increases, but some of the increase was attributed to an increase in the minimum wage, which went into effect last month.

An analyst said bad weather had no major effect on income last month, but the slower rise was caused mainly by particularly strong payments in December. Personal income is an important indicator of the money Americans have available to spend and can be a clue about the economy's future growth.

The January increase in personal income, a measure that can give clues to the nation's economic health, was the smallest since a $4.1 billion rise in January 1977.

The department cited these special factors:

Government payments to wheat" farmers of $700 million in December under the 1977 Food and Agriculture Act contributed substantially to the December figures. In January, these payments added $250 million.

A large year-end divident payment by General Motors Corp. added about $3 billion at an annual rate to dividend payments in December.

The coal strike reduced wages and salaries about $3 billion at an annual rate in December.

Higher Social Security tax payments reduced personal income by about $3 billion at an annual rate.

All the business inventory reduction was at the retail level, the department said. It estimated seasonally adjusted inventories at $332.7 billion in December, down $7 million from November.

The dip of less than 0.1 percent was the first since December a year ago and only the second monthly decline in two years.

The department said sales rose 2.3 percent in December compared with 1.3 percent in November. Total sales were estimated at $235.6 billion in December. For the year, sales were estimated at $2.68 trillion compared with $2.4 trillion in 1976.

Despite the small drop in inventories of December, the department said inventories rose $26.3 billion for the year compared with $24.5 billion in 1976.

Inventories - goods on hand ready for sale or delivery - are an important part of the nation's productive capacity. When inventory accumultion fails to keep pace with sales, job opportunities stagnate or worsen.

The decline in inventory accumulation was more significant in symbolic terms than in actual dollar amounts.

The department said inventories rose $190 million at the manufacturing level and $351 million at wholesale. But retail inventories fell $548 million.

It said retailers reversed their inventory patterns from November when the increased stocks by $1.03 billion. Most of the change took place in clothing, food, energy and other nondurable goods where there was a swing of $1.1 billion.

Wholesalers and manufacturers slowed their inventory growth. Wholesale inventories of durables rose $416 million compared with $628 million in November. Wholesalers cut nondurable inventories by $65 million following a $210 million increase.

Manufacturing inventories of durables increased $297 million compared with $644 million in December. Nondurables fell $107 million, an improvement from the $271 million reduction of November.