A typical legitimate use of a Shell Corporation would be to obtain early seed funding by investors with enough wealth to speculate on a whim. You are buying a promise of something at such a low price that if you were wrong you wouldn't care. It's akin to regular people buying a few lottery tickets and not minding when they become worthless. The promise is that eventually, the ownership of shares in the shell company will translate into real ownership of shares in the real company which the shell was created for. To be more abstract though, Speculators looking to diversify their investment portfolio just want to put money into a genre of a company (assume medical), so Wall Street types will make a shell company with an abstract name that sounds medical and investors will park money in that. Those same Wall Street type people will now seek a non-public medical company that has some promise and try to sell their shell company to them in order to finish the attachment. To clean up later, the listing agencies will get a notification of a ticker symbol and/or company name change and Voila! the shell company becomes a real company.
However, suppose you were a nefarious person on Wall Street and you wanted to get a new Lamborghini but you basically didn't have enough because you spent yourself out on mansions and yachts. You could launch a shell company with the following plan to fleece the general public.
In the past, you would get an accomplice but with technology being what it is, you might be able to go it alone. One thing that makes a company look good is if it's price steadily increases over time. A shell company—unlike a high volume publicly traded real company—has its share price determined by very few people who know about it. With this shell, you and your partner can buy and sell your shares to each other on the open market. Nobody will be paying attention to your trades because it's just a shell company. Every morning you lay out your plans of how you are going to price the trades throughout the day and what volumes you are going to trade at to look just like an ordinary good company when charted. Each day is about the same as the last but a small amount of price rise is apparent in the chart.
After you and your partner build up some history, you begin loud-talking about the company in public places. This makes people overhearing you think they are getting private information that they can capitalize on. You also start to hype the breakthrough the company has made in technology or medicine through some public means like a shell website for the shell company. When people start searching for more on the new company they overheard about in a bar or restaurant, they find confirming evidence that its gonna be a great thing.
Human nature makes those people promote the finding to friends and relatives because a secret is something you tell one person at a time. The hype starts to grow like a virus and real investors begin to seek out this fictional company to put their real money into it.
You and your partner start to see volume numbers in the trades that aren't your own and you know that other people are buying into your scam. At this point, you can increase the hype for the company and keep the internal trades going to make sure the price is rising. As real people buy your shares, the money they paid is basically yours now. You just keep running the scam until you've sold enough of the shares to buy the Lamborghini, you've sold off all the shares to the general public, or something goes wrong with the ruse. As long as you can avoid the FCC, SEC and other law enforcement agencies as well as any civil charges that are going to look for you, you get to keep all the money.