A $1.08 billion unsolicited offer for The Singer Co. was made yesterday by Florida investor Paul A. Bilzerian, who disclosed last Thursday that an investor group he leads had purchased nearly 10 percent of the aerospace company's stock.

The offer was the first significant hostile acquisition effort since the stock market plunged two weeks ago. Some analysts said after the plunge that it would be likely to slow the rate of mergers and acquisitions in the United States, particularly those financed with high-yield "junk" bonds.

Bilzerian, who has been unsuccessful in several earlier takeover attempts, said in a telephone interview that his proposed acquisition of Singer would not involve any high-yield bond financing, and he predicted that a flurry of acquisition activity will develop in the next three months. The combination of lower share prices and lower interest rates makes hostile bids more attractive, he said.

Bilzerian said that the acquisition of Singer would be financed with $100 million put up by his partnership, $150 million from the sale of preferred stock to an affiliated partnership, $355 million in senior preferred shares that Shearson Lehman Bros. has already said that it will buy and $540 million of senior bank debt.

With his $50-a-share offer, Bilzerian is offering a substantial premium over the recent low price of below $30 a share. However, Singer shares closed yesterday on the New Stock Exchange at $47.25, up $3.125.

Singer advised shareholders yesterday to wait until it completes a review of the offer by Nov. 16.

Bilzerian also said in the interview that if he is successful in acquiring the Montvale, N.J., company, he would consider selling Singer's defense business, which accounted for 80 percent of the company's $1.7 billion in 1986 revenue.