We're considering Civicrm for our membership database, but my client would like to see an example of what the look/feel would be for a new member signup. Is there an example/demo that someone can point me to?
There is the generic CiviCRM demo, but that only shows the back-office part (see https://civicrm.org/demo). I assume your client would be interested in the outside facing web part, and that entitely depends on what you can do with Drupal, Wordpress or Joomla. It will be hard to show you one example that gives you an idea of what the new member sign up looks like not because they are not there but because there are so many!
Logged out you can see an example membership page here but what you would see would entirely depend on your theming and there is much that can be done via js to manipulate the page to look very different.
A couple of more extreme examples are here (NZ Greens) and here (Aus Greens) (actually the latter is only donation currently but just trying to give you an idea that what comes out of the box, and what you can get built with time/skills/budget is vastly different)