Using Carl Barks and Don Rosa as "official" sources, a history of Scrooge and Goldie runs something like this.  (Click the numbers to see what stories each bit of information is from.)

Scrooge McDuck arrived in Skagway, Alaska, in 1896, from Australia.  He made the hard journey up to Dawson, which had yet to explode into a thriving boom town.  One of the only things that existed in Dawson at that time was the Blackjack Saloon, run by a certain blonde duck named Goldie O'Gilt.  Scrooge passed by without stopping, but Goldie invited him in for a meal.  He gruffly declined and continued on his way, leaving behind an intrigued Goldie.³

After establishing his claim at White Agony creek, Scrooge returned to Dawson for building supplies.  By this time the influx of people attracted by the Gold Rush had allowed Goldie to expand her business.  As Scrooge passed by, she called, "Hey, handsome!  Come back in a few weeks for the grand opening of the new Blackjack Ballroom!"  Scrooge again flatly refused. ³

Later, Scrooge came back to Dawson to file an official claim on his homestead at White Agony Creek.  Soapy Slick, the villainous loan shark, kidnapped Scrooge and chained him up on his gambling barge.  Scrooge caused a considerable amount of havoc while breaking free, and Goldie was secretly impressed.³

Soon thereafter, Scrooge discovered a gargantuan chunk of gold he dubbed "The Goose Egg Nugget".  Finally he decided to enter the Blackjack Ballroom to show off his find. (When he arrived, Goldie was singing "After the Ball.")  Impressed and greedy, Goldie invited him to have coffee with her.  He reluctantly agreed. Goldie secretly drugged his coffee, and Scrooge awoke hours later in a snowdrift outside of town.  Upon discovering all his gold was missing, he returned to the Blackjack Ballroom and started a tremendous fight, taking on the entire bar at once.¹ (The violence of this scene is one of the primary reasons the censors removed this sequence from the original 1953 publication of "Back to the Klondike".)

After winning the fight, Scrooge cornered Goldie and demanded she return his gold.  She angrily hit him on the head with the Good Egg Nugget but couldn't produce the rest of the missing gold.  To teach her the value of hard work, Scrooge essentially kidnapped her and made her work at his claim for a month.¹   (Therein lies the second reason the flashback was cut out from the original edition.  As Carl Barks himself put it, "What did he do with her at night? I had really overstepped the bounds, and I realized it when the editors cut the sequence out."  Perhaps Goldie's stay at White Agony was not completely involuntary, and the implication of a more intimate relationship irked the censors enough to remove the sequence.  The flashback and its missing art were replaced in 1981, and have been included in subsequent reprints.)

When the month was over, Scrooge gave Goldie a meager salary for her labor.  She threw the money in his face and stomped off, claiming she had dug even more gold than he had. ¹

In January 1898, Scrooge made a trip into Dawson for supplies.  By this time, Goldie's friends at the Blackjack Ballroom suspected she had feelings for Scrooge, and teased her about it.  She angrily denied it, of course.  Meanwhile, Colonel Sam Steele, Superintendent of the Northwest Mounted Police, arrived in Dawson to restore law and order.  Scrooge's enemies wanted him locked up, but they were too scared to personally press charges.  Goldie decided it would be a convenient way to see him again, so she pressed charges against him for kidnapping her earlier.  A Mountie named Scarth went to White Agony to arrest Scrooge. 

When Scrooge learned who had pressed charges, he evaded Scarth and rushed to get back to Dawson to talk to Goldie.  A blizzard came up, and Scrooge collapsed from exhaustion.  While unconscious, he hallucinated about Goldie beckoning him.  The dream took on a Christmas theme as Scrooge became half-aware of a reindeer sleigh approaching.  An Icelandic man had gotten lost on the way to Dawson, but Scrooge showed him the way in exchange for a ride.  They arrived just as Dawson started to go up in flames.  Racing against fire and Mounties, Scrooge ran to the Blackjack Ballroom.  Goldie was waiting for him in the burning building, silently daring him to rescue her. When he stubbornly refused to make a move, she pretended to faint.  Before he could reach her, however, he was knocked out by a chunk of ice from a fire hose.  Goldie ended up saving him from the fire, tried to make it look like he had rescued her, but was taken away by the Red Cross before he awoke. 

The misunderstanding with the law was cleared up, and Scrooge was free to head back to his claim.  Goldie asked the Mounties to deliver a letter to Scrooge on her behalf.  The letter was implied to have been an apology and confession of affection, but Scrooge figured it was more insults and left it in the snow, unread.⁶

Scrooge must have realized the true nature of his own feelings, however, because he later made plans to visit Goldie under better circumstances.  By this time, he had amassed a million dollars, and was ready to leave the Klondike in triumph.  After closing up his claim on White Agony Creek, he started out for Dawson with what was later revealed to be a box of chocolates and a love letter to Goldie.  (And quite possibly a marriage proposal as well!)  Fate stepped in, however, and Scrooge lost his sled deep in a fissure of Mooseneck Glacier.  To make things even worse, he was then attacked by wolves and fell off a cliff, miraculously landing on one of Soapy Slick's gambling barges instead of the icy river.  Scrooge was briefly crestfallen at the de-railing of his plans, but then resolved to dedicate his life to enlarging his fortune.  He bought the bank in Whitehorse, then expanded into many other areas of business.²

In five years, Scrooge was a billionaire, and returned briefly home to Scotland.  While discussing the land he purchased in Calisota, his two sisters noticed a lock of gold hair in his safe-deposit box.  They teased him about having a girlfriend, angering him and causing him to make some Freudian slips, such as "Goldiesota" and "Goldieburg" instead of Calisota and Duckburg.  Finally, Scrooge became furious and yelled at his sisters to shut up.  The matter was more or less dropped after that.⁴

Goldie and Scrooge had no known contact for many decades, although he did reminisce about her privately on Christmas Day, 1947.⁵

Some years later, after medical treatment for memory loss, Scrooge suddenly "remembered" his years in the Klondike, specifically a secret cache of gold he never recovered.  He took his nephews north with him to retrieve the lost gold.  Along the way, he briefly told them about Glittering Goldie.  He said wistfully, "She was spangled and flashy.  The only live one I ever knew!"  When the nephews pressed him for more information, he explained away his wistful attitude by saying Goldie owed him money. The gold she stole from him was worth $1,000, which, with compound interest, amounted to a billion dollars!
Upon reaching Dawson, Scrooge visited the old Blackjack Ballroom, now fallen into disrepair.  When Huey, Louie and Dewey demanded more details about Goldie, he told them about the fight and Goldie's time at his claim.  (Perhaps he left out some important details because of the boys' young age? ;-)

The next day, Scrooge and his nephews went to White Agony Creek to recover the gold cache.  Upon reaching the claim, however, they were shot at by an unseen claimjumper!  After dealing with a guard bear, they discovered that the claim jumper was none other than Goldie herself!  Scrooge was pleasantly surprised, in a way, but tried not to show it.  After all, he just wanted his money from her, nothing more--right?

She tried to distract him from the debt by flirting, but it didn't work; Scrooge demanded payment in full.  Of course she didn't have a billion dollars.  All she was able to pay was the deed to the claim and her last jewelry.  When Scrooge asked where the rest of her money went, she said she had used it to take care of orphans.  She then walked off on her way to the poorhouse.  Scrooge's conscience got the better of him, however, and he offered to forget the debt if she could dig more gold than he did.  Unbeknownst to his nephews, he rigged the contest so that Goldie would find his secret cache.  Overjoyed, Goldie was able to stay on the claim as she had since the end of the gold rush, plus keep all the gold.  Scrooge made a great show of disgust, pretending he had forgotten to take his memory medicine.  His nephews counted his pills, however, and figured out what he had done, but presumably kept the knowledge to themselves.¹

A few years later, a telegram arrived from Scrooge's bank in Whitehorse, informing him that the sled he lost in Mooseneck Glacier all those years ago was about to break free.  Once again he and his nephews went to the Yukon.  When they arrived in Dawson, they were surprised to see that the old Blackjack Ballroom had been restored and made into a hotel, The Frostbit Arms.  Inside, they learned who the owner was: Glittering Goldie.  She had used the gold gained in their last meeting to pay for the restoration.  Scrooge was very uncomfortable  with the situation, but Goldie offered him a free room "for old time's sake", and he never passes up anything free!  In the morning, Donald accidentally told Soapy Slick, Scrooge's old nemesis, the reason for their visit.  Thinking the lost sled contained something extremely valuable, Soapy took his last gambling barge up the river toward the glacier.  Despite insults from Scrooge, Goldie lent him her hot air balloon.  In it, Scrooge and his nephews were able to just barely beat Soapy to the glacier.  They ended up riding a gigantic chunk of ice down the river all the way to Dawson.  When the mini-glacier crashed ashore, the sled broke free.  Inside were all Scrooge's old mining tools, his tattered clothes, and other momentos from his life in the Klondike.  The box of chocolates Goldie was supposed to have received 50 years ago had fallen off in the street.  Huey, Dewey and Louie delivered it to its rightful owner.  Goldie looked flattered and surprised by the letter within, but never commented on what it said.  Once again Scrooge and Goldie parted. ²

The most recent documented meeting between the two took place a few days after Scrooge's Golden Jubilee, the 50th anniversary of his arrival in Duckburg.  As a special surprise, Scrooge's friends and family paid to fly Goldie to Duckburg.  It was indeed a surprise, and most certainly special.  She showed up at the Money Bin, gave Scrooge her gift--a passionate kiss--and left.  Their parting words, I think, sum up their relationship.  Scrooge said, "I have a worldwide empire to run!  I can't just drop everything and fritter away time on nonsense!"  Goldie replied, "I know!  I know!  I'll wait."

And to the best of anyone's knowledge, she still is.  Nevertheless, Scrooge's irritation quickly turned to joy when she was out of sight.⁷

Later, Scrooge went to great lengths to retrieve a particular coin when it went A.W.O.L. from the money bin. Unbeknownst to Donald and the others, the coin was one of the ones Scrooge had attempted to pay Goldie with after her time at his White Agony claim. She, of course, had thrown the money at him angrily and left, so the coin remained a special part of his own collection.⁸

Scrooge, Donald, Huey, Dewey and Louie once journeyed to Finland in search of the legendary Sampo, a magic machine capable of cranking out unlimited quantities of gold coins (and salt and wheat, if you get hungry!)  After an epic battle with Magica DeSpell and the magical characters of the Kalevala legend, Scrooge clung to the Sampo machine as it ascended to Eternity with the good wizard Väinämöinen.  In return for his help in reconstructing the Sampo, Väinämöinen gave Scrooge permission to come with him, to eternal prosperity.  The catch, however, was that Scrooge could never return to a normal, mortal life.  Väinämöinen warned him that if he came along, he would never again see "the grandeur of the northland", where a "lost love still awaits" him.  Gee, I can't imagine who he was referring to...  :-)  With a distant look, Scrooge decided that, no, nothing was worth that high a price.   Väinämöinen congratulated him on choosing correctly, then let him fall back to Earth, still grasping the handle of the Sampo. One of the nephews said Väinämöinen let him keep it for his trophy room, but Scrooge wistfully said it was to remind him of why he returned.  When Donald questioned him, he changed the subject.^{9
(Now even supernatural beings are telling him to go back for Goldie!  How many hints does he need? ;-)}

And there the story stands...for now.

What about Paperetta/Dickie?