The noun garbageology is from garbage and the combining form -ology, denoting a particular subject or science.

This noun was originally used humorously to denote the academic study of refuse collection—the following is from the column Bats in the Belfry, by ‘The Gangrel’, published in the Daily Record and Mail (Glasgow, Lanarkshire, Scotland) of Thursday 9^th April 1942:

The Pincush Utopia
CAPTIOUS critics are still yapping at the heels of the great thinker and super-economist.
Writes one:—
“The Pincush Plan gives everybody £5000 a year and jails us if we don’t spend it. To get rid of the money we have to go to work, paying for the privilege of choosing an occupation. Okay! Won’t everybody choose the easy and agreeable jobs, like writing a newspaper column or negotiating with Gandhi? Will anyone choose to be, say, a dustman?”
No Bother Whatever
DR. PINCUSH replies:—
“Under my scheme it will cost practically nothing to get an enjoyable job, therefore this will be of no real help in spending one’s £5000. Disagreeable jobs, on the other hand, will be sold at very stiff prices and will be much in demand.
“Moreover, these unpleasant occupations will be glamourised by the Ministry of Information. Refuse collection, for example, will be known as serendipity. There will be broadcast talks and pamphlets on the romance of rubbish and ashcan allure. Most Universities will have a Chair of Garbageology.”

—Cf. also ‘garbology’: original meanings and oddities.

The noun garbageologist is from garbage and the combining form -ologist, denoting a person who is concerned with a particular subject or science.

This noun was originally used humorously to designate a person employed to collect domestic refuse. The earliest occurrences that I have found are as follows:

1-: From one of the unconnected paragraphs making up the column Editorial Points, published in the Boston Evening Globe (Boston, Massachusetts, USA) of Thursday 1^st August 1940—the noun occurs as garbagologist:

Some day the rest of us will wake up, and we’ll have to call the swill man the garbagologist.

2-: From the column Good Afternoon, by Ray Redmond and John Jennrich, published in the Akron Beacon Journal (Akron, Ohio, USA) of Tuesday 21^st January 1969:

Experts In Garbage
For a long time, school janitors have been called maintenance engineers. Now the garbage man has a new title. He was hailed as “your friendly garbage-ologist” by Streetsboro Councilman Robert French the other day.

—Cf. also ‘garbologist’: original meanings and early occurrences.

The noun garbageologist came to also designate an expert in the recycling of waste materials. The earliest occurrences that I have found are as follows:

1-: From the following story, by James R. Campbell, of United Press International, published in several U.S. newspapers on Sunday 14^th June 1970—for example in The Palm Beach Post-Times (West Palm Beach, Florida):

OKLAHOMA CITY (UPI)—Those pungent zephyrs from the city dump are beginning to smell sweet to some people.
These are people who know trash piles can supply new raw materials for industry, as well as energy to drive the gears and burn the lights.
“A lot of big blue chip companies smell money in garbage,” said Oklahoma City industrialist H. Dale Jordan.
[…]
It is not only the major companies that have turned their attention to the hot topic of ecology.
Henry Munde, Southwest Factories vice president, spends most of his time on the solid waste problem. He jokingly refers to himself as the country’s “foremost garbageologist.”

2-: From the portrait of one of the persons who were likely to succeed James V. Segreto as county counsel, published in The Record (Hackensack, New Jersey, USA) of Sunday 20^th September 1970:

JOE CARUSO, described by his friend, former Passaic Valley Water Commissioner Stanley M. Levine, as a garbageologist, says no matter how much fun you poke at his profession the fact remains those that practice it are doing more than their share to promote the ecology.

3-: From The Xenia Daily Gazette (Xenia, Ohio, USA) of Monday 19^th April 1971:

‘ECO-JOY’
Waste recycling methods to be demonstrated at YS fair

YELLOW SPRINGS—Noted “garbageologist” Dr. Geoffrey Stanford has formulated plans for the complete recycling of waste materials and has interested the Village Council here into trying his proposals on a small scale.
Dr. Stanford plans to demonstrate his waste recycling methods at the April 25 “Good Earth Fair,” at the Bryan Community Center.
His students at Antioch College call Dr. Stanford a garbageologist or the nomer [?] he prefers, “peopleologist.”
“Waste disposal is one of my prime concerns now,” says the spry surgeon, photographer, optical scientist, director of medical, industrial and educational films and city planner, “. . . but people are my prime interest all of the time.”

The noun garbageologist came to also designate a person who looks through the garbage of celebrities, politicians, etc., especially in search of compromising or incriminating material. And the noun garbageology came to also denote the practice of looking through the garbage of celebrities, politicians, etc. The earliest occurrences that I have found are from the Deseret News (Salt Lake City, Utah, USA) of Friday 3^rd December 1971:

That’s A Lot Of Rubbish!
By NICK SNOW
Deseret News Staff Writer

Can prowling through a celebrity’s trash cans catapult an ambitious freak to fame on the pages of a national magazine? If the freak’s name is A. J. Weberman and the trash cans belong to Bob Dylan, ‘garbageology’ is a gold mine and there is always somebody ready to believe that “you are what you throw away.”
An avid “Dylanologist” to begin with, Weberman says he hit upon the garbage can idea when “the Howard Hughes of rock” refused to see him about a year ago. “He knew I’d been studying his poetry for years, that I had more insight into it than any other rock critic. Did (Samuel) Johnson throw (James) Boswell out? What a lot of garbage . . . ‘Garbage’. The light bulb went on.”
Weberman went to the service entrance of Dylan’s New York apartment with his wife. “I reached into the plastic can and pulled out a crumpled sheet of paper. It was a draft of a note to Johnny Cash. ‘Ann,’ I said, ‘this ain’t no garbage pail, it’s a gold mine.’”
The initial find prompted him to return. “I began to make regular pickups of Dylan’s garbage each night. I never thought I’d stoop so low, literally. […] There were invitations to attend special sales at exclusive department stores along with all kinds of fashion magazines addressed to Sara.” Dylan’s wife, claims the “garbageologist,” “seems to be into a plastic ‘manakin’ trip.”
[…]
By March he had enough for an article which appeared in various underground newspapers where it caught the attention of the editors of Esquire magazine, who contacted Weberman and asked him to apply his science to other celebrity garbage cans. Weberman did; and the November issue of the magazine carries information on the garbage of Yippie leader Abbie Hoffman, playwright Neil Simon and prizefighter Muhhammed [sic] Ali as well as Bob Dylan.
“I admit that the image of a long-haired yippie sifting through trash in the middle of the night might strike some as sort of weird,” Weberman concedes, “but garbageology is a great way to find out what people are really like.”