List column value for all databases that have a specific table and column
I really, really, really recommend against sp_MSforeachdb. It's easy, sure, but it is undocumented, unsupported, and buggy as all get out.

Here is a dynamic SQL solution:

CREATE TABLE #x([Database] sysname, ApplicationVersion varchar(32));

DECLARE @sql nvarchar(max) = N'';

SELECT @sql += N'
  BEGIN TRY
    INSERT #x EXEC ' + QUOTENAME(name) +N'.sys.sp_executesql 
    N''SELECT DB_NAME(), MAX(Version) FROM dbo.ApplicationVersion'';
  END TRY
  BEGIN CATCH
    PRINT CHAR(32);
  END CATCH;'
FROM sys.databases 
WHERE state = 0; -- database is online

EXEC sys.sp_executesql @sql;

SELECT [Database], ApplicationVersion FROM #x;

DROP TABLE #x;