Which design pattern do you think is the most popular?
I'm pretty sure the most common is 'The Big Ball of Mud'. Unfortunately for us all.
If you're talking about Gang-of-Four design patterns I would have to say Singleton. This is a really sad state of affairs - almost as if programmers hear about the magic of design patterns and then simply stop after the first one.
If you're talking about architectural-style patterns (in other words, design patterns which span multiple classes or layers) I would have to say MVC. A few other popular ones are the repository pattern and service-locator.
Think of any REST interface like Twitter or the SO APIs. It's hiding a bunch of back end stuff that, at least in twitters case, can be horribly complex.
Do I care which db tables @apklusk's tweets are stored in? And that it has a fast cache? And that @shemnon has such low readership the tweets are always out of memory? That's the magic of the Facade.
Along with my previous answer, I'm going to add a new one I ran into today from my beast of an inherited project:
While debugging I ran into a piece of code calling 2 similarly named functions:
if(x) Func1 else Func2
Func1 and Func2 had the same parameters. I pulled them out and diffed, and they are 100% the same except for Func1 calls Func3 and Func2 calls Func4. I pulled those out and diffed them. 100% the same, except for 1 line. So a previous programmer decided to copy and paste two 50-line-nested functions, instead of adding a flag parameter, to 2 private functions used in exactly 1 place. It makes me want to cry.
I don't think it's listed in the GoF, but I use variations of it all the time.
The most frequently used design pattern is, obviously, Spaghetti.
At the software design level, I would argue that Composite, Strategy and Template (and to a lesser extend, Factory) are the most widely usage. For the last decade or so we have seen a preference for composition over inheritance when building our systems using a OO design. As a result, combination of the patterns just mentioned play a much more important role than all others IMO.
Architecturally, I'd argue for MVC on the enterprise, and patterns based on state machines on the embedded side of things.