SQL Azure and Full-Text Catalogs, Indexes and Searching
SQL Azure Database does not currently (January 2014) support Full Text Search. Many requiring this functionality have opted to use Lucene.NET instead. There's a useful overview of Lucene.NET plus code samples below:
Lucene.NET Overview and Code Samples
EDIT - October 2015:
Full-Text Search is now available in Azure SQL Database (GA)
As of August 2014, Microsoft is offering a Search-as-a-Service named Azure Search:
Azure Search is a search-as-a-service solution that allows developersand cloud architects to embed a sophisticated search experience intoweb and mobile applications without having to worry about thecomplexities of full-text search and without having to deploy,maintain, or manage any infrastructure.
With Azure Search, a developercan enable the power of searching data in their application, reducethe complexity around managing and tuning a search index, and get upand running quickly using familiar interfaces and a consistentplatform.
See http://azure.microsoft.com/en-us/services/search
Update:
As of April 30, 2015 - Full Text Search is available in Preview for SQL Azure. See http://azure.microsoft.com/blog/2015/04/30/full-text-search-is-now-available-for-preview-in-azure-sql-database/
Looks like they recently added the feature: