PROBLEM: I have to show the sport events of my softball team to my mom.
SOLUTION: Users should have a permanent address like http://calendar.google.com/username/calendarname that displays public events in a human-readable form (like PHP iCalendar does). You could share this URL to people that don't use Google Calendar.
PROBLEM: I've created a calendar of great cultural events in New York. How can the visitors of my site grab the calendar to their accounts without copy-paste?
SOLUTION: Google should provide an address like http://calendar.google.com/addcalendar?url=calendar_address you can put on a site, so visitors can easily add your calendar.
PROBLEM: You like to search events with Google Toolbar or you don't want to log in to Google Calendar just to search next week's program of the Met.
SOLUTION: You should search public calendars using a simple link like http://calendar.google.com/search?q=query and output the results as a new calendar.
PROBLEM: I want to copy Italy's matches from FIFA World Cup 2006 Calendar to my calendar.
SOLUTION: There should be an easy way to mark certain events in a calendar for copying or deletion.
PROBLEM: I want to use my favourite feed reader to check today's agenda.
SOLUTION: The feeds should have parameters to sort the events by date, to show only the current events, to restrict the events to an interval. If the feed would show only the current events, you could use it the same way you check mail with Gmail feed.
Popular Google Calendars
Google Calendar module for Personalized Homepage
Google Calendar plugin for Google Desktop