Usually introduced by the definite article, the phrase singing milkshake, also Singing Milkshake, is a nickname that has been given, in particular, to the British-Australian singer and actress Olivia Newton-John (born 1948).

A student named Tony Gasparre explained the metaphor in the review of Sideroxylon, the debut album of the Sydney rock band The Celibate Rifles, published in Tharunka (Kensington, New South Wales) of Tuesday 7^th June 1983 [erroneously dated Tuesday 7^th June 1999]—Tharunka is a student magazine published at the University of New South Wales:

You can compare most popular music around with a milkshake. It gives you three minutes of pleasure and then you throw it away. You can even get singing milkshakes like Olivia Newton-John. The laws of milkshakes music are that the lyrics should not be too challenging and the music not too inspiring.

These are the earliest occurrences of singing milkshake that I have found, in chronological order:

1-: From an interview of Helen Mirren ¹, by John J. Archibald, published in the St. Louis Post-Dispatch (St. Louis, Missouri) of Tuesday 27^th February 1979:

Contemporary music, it developed, is one of Helen Mirren’s great interests.
“There aren’t many outdoor concerts anymore,” she said. “Too much trouble with the crowds, I suppose. But there are many small clubs and theaters in London where the great artists play.”
Miss Mirren didn’t hesitate when asked for some of her favorites.
“Ian Drury . . . Bob Marley . . . Muddy Waters . . . Randy Newman . . . Kate Bush . . . and, of course, Millie Jackson.”
There are some popular American performers who are not to her taste. “I do not like Olivia Newton-John,” she said. “I think of her as a singing milkshake.”

¹ Helen Mirren (Helen Lydia Mironoff – born 1945) is an English actress.

2-: From TV Guide, published in the Liverpool Echo (Liverpool, Merseyside, England) of Monday 19^th April 1982:

BBC2
[…]
8.10 Barry Manilow ² in Britain. Return of the singing milkshake for yet another golden gargle around Britain. The Beeb filmed two of his Royal Albert Hall concerts.

² Barry Manilow (Barry Alan Pincus – born 1943) is a U.S. singer-songwriter.

3-: From Television Choice, by Richard Coleman, published in The Sydney Morning Herald (Sydney, New South Wales) of Monday 14^th February 1983:

Olivia In Concert, 7 at 7.30 pm: No, Lord Olivier has not taken to the microphone. This is Olivia, our very own Olivia, once described by a nasty English journalist as “the singing milkshake”. Tonight she performs in a simulcast on 2DAY-FM so you can have two milkshakes for the price of one.

4-: Cf. the above-quoted review published in Tharunka (Kensington, New South Wales) of Tuesday 7^th June 1983.

5-: From Ode to Samuel Taylor Coleridge and Olivia Newton-John, by ‘Young Doonks’, published in Tharunka (Kensington, New South Wales) of Tuesday 4^th October 1983—this is a parody of Kubla Khan (1816), a poem by the English poet, critic and philosopher Samuel Taylor Coleridge (1772-1834):

In Xanadu ³ was Newton-John
A singing milkshake, looking nice
But not on honey-dew we fed
Nor drunk the milk of Paradise

And e’er she blossomed Physical
So as to get her man,
But midst that sunny Pleasure-dome
She could not live. . .but Kubla Khan

³ Xanadu:
– An idealised place of great or idyllic magnificence and beauty, as portrayed in Coleridge’s poem Kubla Khan.
– The title of a 1980 U.S. musical film starring Olivia Newton-John.