Dropping unnamed constraints

For dropping an individual unnamed default constrain on a column use the following code:

DECLARE @ConstraintName VARCHAR(256)
SET @ConstraintName = (
     SELECT             obj.name
     FROM               sys.columns col 

     LEFT OUTER JOIN    sys.objects obj 
     ON                 obj.object_id = col.default_object_id 
     AND                obj.type = 'F' 

     WHERE              col.object_id = OBJECT_ID('TableName') 
     AND                obj.name IS NOT NULL
     AND                col.name = 'ColunmName'
)   

IF(@ConstraintName IS NOT NULL)
BEGIN
    EXEC ('ALTER TABLE [TableName] DROP CONSTRAINT ['+@ConstraintName+']')
END

If you want to do this for a default column, which is probably more common than the original question and I'm sure a lot of people will land on this from a Google search, then just change the line:

obj.type = 'F'

to

obj.type = 'D'

There is not a built in procedure to accomplish this, but you can build your own using the information in the information_schema views.

Table based example

Create Proc dropFK(@TableName sysname)
as
Begin

Declare @FK sysname
Declare @SQL nvarchar(4000)
Declare crsFK cursor for

select tu.Constraint_Name from 
information_schema.constraint_table_usage TU
LEFT JOIN SYSOBJECTS SO 
ON TU.Constraint_NAME = SO.NAME
where xtype = 'F'
and Table_Name = @TableName
open crsFK
fetch next from crsFK into @FK
While (@@Fetch_Status = 0)
Begin
    Set @SQL = 'Alter table ' + @TableName + ' Drop Constraint ' + @FK
    Print 'Dropping ' + @FK
    exec sp_executesql  @SQL
    fetch next from crsFK into @FK
End
Close crsFK
Deallocate crsFK
End

Although Gunner's answer puts people on the right track if you want to drop an actual DEFAULT constraint rather than an FKey constraint (which is what brought ME here too!) there are problems with it.

I think this fixes them all. (T-SQL)

CREATE PROC #DropDefaultConstraint @SchemaName sysname, @TableName sysname, @ColumnName sysname
AS
BEGIN
    DECLARE @ConstraintName sysname;

    SELECT @SchemaName = QUOTENAME(@SchemaName)
         , @TableName  = QUOTENAME(@TableName);

    SELECT @ConstraintName = QUOTENAME(o.name)
      FROM sys.columns c 
      JOIN sys.objects o 
        ON o.object_id = c.default_object_id 
     WHERE c.object_id = OBJECT_ID(@SchemaName+'.'+@TableName) 
       AND c.name = @ColumnName;

    IF @ConstraintName IS NOT NULL
        EXEC ('ALTER TABLE ' + @SchemaName + '.' + @TableName + ' DROP CONSTRAINT ' + @ConstraintName + '');
END