If there's a deflation, the advertising and public relations industry will nearly disappear in my opinion.
It didn't disappear during the last depression. Granted, the industry was nothing compared to what it was during the 1920's, but as long as anyone is selling anything, you're going to have advertisements.
I was reading a speech by the new head of the Wisconsin Dept of Tourism that I think was right on target. The gist of part of it was in an era of tight budgets, how do we get the word out that Wisconsin is a great place to vacation. Her solution was that, whenever possible, she needs to get the word out without paying for it, through free interviews or whatever means.
Another part of the equation is going to depend on who takes market share as the Depression deepens. Private label is known for not advertising to keep costs down and probably the question is how much market share they take. The same would be partly true of local manufacturers.
Having worked for a global consumer products manufacturer (Frito-Lay), I knew how much they were spending on advertising and every component of their cost structure. As private label and local manufacturers take market share, advertising is the area where global consumer products manufacturers are going to have to cut in order to survive. They won't have an option.
As far as other industries that rely heavily on advertising, I have no expertise to speak of.
Thinking of it from the macro point of view, it'll depend on how deep the trend against globalization goes and how local our lives become. Part of my assumption in making those statements is that the trend against globalization will go very deep, deeper than it did during the Great Depression. That's only a guess though and may not be correct.
One last point I should add to close the loop. In the Internet/computer era, any bozo who has some time on their hands should be able to pick up the formula for something like Tide laundry soap either by finding a contact who knows a lot about laundry detergents or by finding a contact who has the formula and will sell it. I had a job once where I had access to every formula for every product that a major consumer products manufacturer had on the store shelf. I had it because the government required them to divulge their formulas, which was not required during the Depression. That's just one avenue. Another would be hacking, which also didn't exist during the Great Depression. Once someone has a formula like that, they can mix the product and take it around to flea markets, put it in bulk areas of local businesses, advertise it on craigslist and so on.