AND MANY THAT AREN'T

I don't know where Trump gets his picks for appointments.  Trump's nominations that were withdrawn are too numerous to list here (more than 60) but can be seen at https://en.wikipedia.org/wiki/List_of_Donald_Trump_nominees_who_have_withdrawn
But the most notable of these is Andrew Pudzer who was nominated for Secretary of Labor.  Also, 12 nominations for judge have been withdrawn.

It seems like Trump searches around until the worst possible candidate is found.  Some of these appointments view their office as their private candy store and see how much candy they can force down.  One of these was Tom Price, Secretary of Health and Human Services who "Resigned following scrutiny of his use of private charters and military aircraft for travel.  His tenure was the shortest in the office's history."*

Another candy store Secretary was Ryan Zinke (Sec. Interior) who rode to work his first day in office astride a horse and liked to have a flag-waving when he was in office.  But more seriously of the many investigations Zinke "Like other top administration officials, Zinke has come under fire for his use of private charter planes and government helicopters, including a $6,250 round-trip helicopter flight to meet Vice President Mike Pence for an afternoon horseback ride. "**

Also on Zinke: As the Huffington Post first reported in January, Zinke failed to report that he was still a shareholder in PROOF Research, a Montana company that manufactures and sells firearms. In addition to the undisclosed financial interest, Zinke’s calendar shows he met with executives and a lobbyist for the company — a move ethics experts said may have been a conflict of interest.
For more on Zinke see the reference **

Then there was Scott Pruitt Sec. of the Environmental Protection Agency who leased an apartment from the wife (a health lobbyist) of an energy lobbyist.  He should have known something wasn't right when he was charged only $50.night.  And he even fell behind payments on that. Asked by Fox whether renting a room from a Washington lobbyist was inconsistent with Trump’s promise to drain the swamp, Pruitt replied: “I don’t even think that that’s even remotely fair to ask that question.”

Pruitt was also in trouble with his spending and travel charges, among others.  Actually, Pruitt was against the mission of EPA and filed 14 suits against it when he was Attorney General of Oklahoma.  Four of the suits were against Clean Power.

At the top of the list of terrible appointments must be Michael Flynn who was appointed as Trump's national security officer even though he was an employee of the Turkish government.  Then there was Sebastian Gorka who couldn't get a security clearance.  Rob Porter who had to quit as White House Staff secretary after reports of spousal abuse by two former wives.

Brenda Fitzgerald, Director of the Centers for Disease Control and Prevention, "Resigned due to scrutiny of her financial holdings, which included stock in Japan Tobacco. Her tenure was the shortest in the office's history, excluding interim appointments."*

The most recent departure is Alexander Acosta (Sec. Labor) for giving a sweetheart deal to Jeff Epstein on sexual abuse charges.  You have to wonder why he ever got appointed in the first place.  K.T. McFarland withdrew because she "wasn't a good fit at NAC."  After a few months in office, she was nominated to be ambassador to Singapore, but, eventually, even that offer was withdrawn.

CONTINUING IN OFFICE ARE:

Perhaps the worst case of a Cabinet Officer remaining in office is Wilbur Ross, Secretary of Commerce: The whole story is worth your time. But to highlight, Ross spent most of 2017 in office while maintaining partial ownership of, among other things:
Chinese state-owned enterprises
A shipping company tied to Russian oligarchs
A Cypriot bank that’s involved in Robert Mueller’s investigation
A major player in the auto parts industry with a direct stake in Commerce’s trade policy decisions
Perhaps the most egregiously corrupt aspect of this is that, according to Alexander, Ross even managed to find a way to profit off the exposure of his own corruption: “five days before reports surfaced last fall that Ross was connected to cronies of Vladimir Putin through a shipping firm called Navigator Holdings, the secretary of commerce, who likely knew about the reporting, shorted stock in the Kremlin-linked company, positioning himself to make money on the investment when share prices dropped.”***  He has also lied in inflating his net worth to $3.9 billion whereas financial disclosure statement puts it at $700 million.****

Secretary of Housing and Urban Development, Ben Carson, ordered $31,000 of furniture for his office by the government.  When this turned into a scandal, the ordered was rescinded.  Carson is a famous retired surgeon and must be worth millions of dollars.  Why didn't he just pay for the furniture?  If he didn't want it when he left office, he could have given it to the government.  Carson is a big disappointment for me.   He continues to remain in office.

Rick Perry, Secretary of Energy, wanted to eliminate the Department when he was running for president.  It turns out he didn't know what the Department does (not unusual for political appointments).  When he found out, he changed his mind.

Betsy DeVos, Secretary of Education, was barely approved with the vice-President casting the deciding vote.

Note Added August 3, 2019:  John Ratcliffe withdrew his nomination as director of DNI (Director of National Intelligence) because of lack of qualifications on August 2nd.  Ratcliffe was the guy who told Mueller that part II of the Mueller report was not authorized.

* https://en.wikipedia.org/wiki/List_of_Trump_administration_dismissals_and_resignations
** https://grist.org/article/scandals-pile-up-for-interior-chief-ryan-zinke.
https://www.politico.com/story/2018/04/05/scott-pruitt-condo-rental-504603
*** https://www.vox.com/2018/6/20/17479170/wilbur-ross-corruption
http://stopcontinentaldrift.blogspot.com/2019/01/let-them-eat-cake.html
**** https://www.vox.com/2018/6/20/17479170/wilbur-ross-corruption