While there are many unwritten rules, here are few which I find most significant. To me, these are qualities of any professional web application.
There should be no single point of failure
The site should never be sluggish
The site should never show any errors
If the site is not available there needs to be a maintenance message
It needs to employ recent and update to date technologies
It should be easily maintainable
It should be easily expanded
It should be well documented
It should be have an underlying coding convention
It should be user-friendly
Website layout and design needs to be consistent
I expect nothing less from any modern day professional web application and applications I develop. The applications I develop might not always fulfill these requirements, but it’s definitely something I strive for each time a line of code pours from these hands of mine.