We considered trelloapp.com (and use trelloapp on Twitter), and my guess is that it all would have turned out fine either way. But I still have a possibly irrational attachment to the [name].com
In the mobile world the {product}app.com convention is pretty prevalent and has opened up some choices. You could always start a new convention for web-based products.
I personally don't think a single product needs to have its own unique domain. Most browsers nowadays have the combo URL/search field feature (or the search field is right next to the URL). Someone typing in the product name will likely find it one way or another.
I know of a little anecdotal evidence: A .org I know of used to routinely have to remind people it was .org, not .com. The owner of another .org I know of once stated bluntly they "farked up" by not buying the .com version as well, which was bought at some point by a company (using the same name, if I recall correctly -- physically located elsewhere). And, of course, there is the infamous whitehouse .com to take advantage of the fact that even for .gov sites, people mentally default to the .com mode.
No formal studies or anything (that I know of), but there is some anecdotal evidence that it makes a difference of some sort.