The query below lists all unique indexes in the database.
Check out also list of unique constraints.
Query
select i.[name] as index_name,
    substring(column_names, 1, len(column_names)-1) as [columns],
    case when i.[type] = 1 then 'Clustered unique index'
        when i.type = 2 then 'Unique index'
        end as index_type,
    schema_name(t.schema_id) + '.' + t.[name] as table_view, 
    case when t.[type] = 'U' then 'Table'
        when t.[type] = 'V' then 'View'
        end as [object_type],
    case when c.[type] = 'PK' then 'Primary key'
        when c.[type] = 'UQ' then 'Unique constraint'
        end as constraint_type,
    c.[name] as constraint_name
from sys.objects as t
    left outer join sys.indexes as i
        on t.object_id = i.object_id
    left outer join sys.key_constraints as c
        on i.object_id = c.parent_object_id
        and i.index_id = c.unique_index_id
   cross apply (select col.[name] + ', '
                from sys.index_columns as ic
                    inner join sys.columns as col
                        on ic.object_id = col.object_id
                        and ic.column_id = col.column_id
                where ic.object_id = t.object_id
                    and ic.index_id = i.index_id
                        order by col.column_id
                        for xml path ('') ) as D (column_names)
where is_unique = 1
and t.is_ms_shipped <> 1
order by i.[name]
Columns
- index_name - name of the index
- columns - index columns separated with ","
- index_type - type of the index
- Clustered unique index
- Unique index - non-clustered unique index
 
- table_view - schema and name of the table or view
- object_type - type of object which constraint/index was created on
- Table
- View
 
- constraint_type -
- Primary key - for primary keys
- Unique constraint - for constraints created with CONSTRAINT UNIQUE statement
 
- constraint_name - primary/unique key constraint, null for unique indexes without constraints
Rows
- One row: represents one index
- Scope of rows: all unique indexes in the database
- Ordered by: index name
Sample results
Unique indexes in the AdventureWorksLT database:

 
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                     
                                                                 
                                                             
                                                             
                                                             
                                                             
                                                                 
                                                             
                                                             
                                                                 
                                                             
                                                             
                                                             
                                                             Rene Castro
                                                                        Rene Castro
                                 Azure SQL Database
                                                                Azure SQL Database
                                 SQL Server
                                                                                                SQL Server
                                             Oracle database
                                                                                                Oracle database
                                             IBM Db2
                                                                                                IBM Db2
                                             MySQL
                                                                                                MySQL
                                             MariaDB
                                                                                                MariaDB