As I look at the SmugMug feature comparison chart I see there's an item for "Full screen, silky smooth Flash slideshows". Does SmugMug offer HTML5 support or some other means by which photos can be viewed on iPhones, iPads, or other devices that don't support Adobe Flash?
Yes. If you use the provided homepage slideshow widget, or the in gallery slideshow buttons, then it will "do the right thing" on iOS devices and use an HTML5 based slideshow.
SmugMug has had a mobile theme avaiable for iOS devices since July 13th, 2007. It has gone through a few iterations, but in its current state it still looks very much the same as when it was introduced. It works very well, and even scales just fine to the iPad or iPhone sized screens.
The mobile site is enabled by default I believe, but if you want to switch the setting you will find it here: Control Panel> Settings> Mobile Site> Enabled
Here is an example of what you will see on the mobile site in the menu view:
Slideshows are a bit more tricky then the standard gallery question, and the reason is that really you can customize and setup a slideshow how ever you wish. In years past, we had to find more custom ways to get slideshows embedded into a SmugMug page. Currently, the feature that you are asking in particular about as listed in the feature comparison page is compatible with iOS devices. It is written based on HTML5, a standard that mobile Safari does support. What this means, is that if you are using the standard slideshow feature that SmugMug provides as a homepage option, it will work.
One important thing to note is that the "typical" SmugMug user may actually have two distinct ways of using slideshows on their site. One would be a homepage slideshow to feature the top images that they have. BUT the catch is, to view this you would have to break out of the mobile site, and into the "full site". You can do this by clicking on the link located at the bottom of your mobile site. The second option is a dedicated "Slideshow" button inside of a gallery. This option is essentially another viewing option for that particular gallery.
Example of the Gallery Slideshow option:
If you have an older version of a custom flash slideshow, such as Shizam's Flash Slideshow which is largely outdated at this point - you will not be able to view it on the iOS Safari browser, as it does not support flash currently. This does not mean that all third party slideshows are incompatible, just ones that are based on flash. The very popular JFriend HTML5 slideshow does work on mobile devices for example.
This is an example of a somewhat customized site using the JFriend HTML5 Slideshow. It basically looks the same as the standard one, but with the JFriend option you can customize it to appear "stretchy" for example.
To sum up if you use the standard tools SmugMug provides you can access your galleries and view the site folders as well as the built in homepage slideshows on iOS. If you have a custom slideshow built off of HTML5 or stay within the mobile site, things should look pretty good on iOS as well.
If you use the JFriend HTML5 Slideshow it works just fine on iOS.
The standard "built-in" (via EZCustomizer) slideshow uses Flash, and therefore will not work on non-Flash compatible devices.
You should note that iOS devices will default to mobile view which is optimized for iPhone, in which case your home page slideshow does not load, as you get a Gallery view instead. (this mobile view can be disabled in your control panel).
So to get the slideshow to work on mobile, you will need to use the JFriend HTML5 Slideshow code + disable mobile view in your control panel, or choose the 'Desktop view' link from your mobile phone.