Get the list of unique constraints and indexes in a database

Since unique constraints are implemented under the covers as indexes, you can get all of this information directly from sys.indexes:

SELECT
  [schema] = OBJECT_SCHEMA_NAME([object_id]),
  [table]  = OBJECT_NAME([object_id]),
  [index]  = name, 
  is_unique_constraint,
  is_unique,
  is_primary_key
FROM sys.indexes
-- WHERE [object_id] = OBJECT_ID('dbo.tablename');

To repeat for all databases (and presumably without the filter for a specific table):

DECLARE @sql NVARCHAR(MAX) = N'';

SELECT @sql += 'SELECT db = ' + name + ',
    [schema] = OBJECT_SCHEMA_NAME([object_id]),
    [table]  = OBJECT_NAME([object_id]),
    [index]  = name, 
    is_unique_constraint,
    is_unique,
    is_primary_key
  FROM ' + QUOTENAME(name) + '.sys.indexes;'
FROM sys.databases
WHERE database_id BETWEEN 4 AND 32766;

EXEC sp_executesql @sql;

The other answers did not return complete lists for me. This query worked for me to return all unique indexes that are not primary keys or system tables:

select i.name as index_name, o.name as object_name
from sys.indexes i
    join sys.objects o on i.object_id= o.object_id
where (i.is_unique_constraint = 1 OR i.is_unique = 1) 
    and i.is_primary_key = 0 and o.type_desc <> 'SYSTEM_TABLE'