How professional are the shopping carts available (Miva, osCommerce, etc) that various web hosting companies are offering? What features do they lack that one might want to have? Are they flexible? I know that these questions are a bit broad, but I want to hear peoples general opinions first before asking more specific questions. However, if people feel that I need to be a bit more specific, I will gladly do so.
Also, while we are on the subject, which shopping carts do people recommend and why (note: I wish to emphasize those carts that people recommend and less so those that people don't recommend).
All the shopping carts you listed are "professional" and do what they are designed to do. The trick really is to understand your requirements first, then select the solution that most inherently matches your requirements.
We work with several shopping carts and will recommend one to our clients based on their specific requirements. If your primary concern is SEO and you need to integrate with something like quickbooks or you need to integrate a cart into an existing site, take a look at shopsite. If you're just starting out or have very specific functional requirements, take a look at Zen Cart or osCommerce (as they come pre-loaded with a ton of functionality). If you have a good budget, want something that looks and acts in a very specific manner, you may be better off with a custom solution.
I know you started off very broad for a reason, but, this is a really difficult question to answer unless you provide some specifics around your actual requirements/expectations...