This will find items matching a particular template.
fast://*[@@templateid='{072A5DF2-4283-4043-9A4F-7D1218F4D899}']
Without text in the item name.
fast://*[@@templateid='{0400B345-381D-4BAF-8B83-85DD52632FE5}' and @@name != '%test%']
Created after a certain date.
fast://*[@@templateid='{0400B345-381D-4BAF-8B83-85DD52632FE5}' and @@name != '%test%' and @__Created >= '20150201']
This will find items that have a particular sublayout.
fast://*[@__Renderings='%{D1473911-D4F7-4990-B035-09A2C689BF58}%']
This will find items that a particular custom field value.
fast:/sitecore/content/Bus/#Meta-Data#/Board Members/*[@Board = 'OTM']
Note that in Sitecore 8.x the XPath Builder can be found at /sitecore/shell/default.aspx?xmlcontrol=IDE.XPath.Builder
It has been removed in 9.x+.