Adding a new SQL column with a default value

Try this:

ALTER TABLE table1 ADD COLUMN foo INT DEFAULT 0;

---

From the documentation that you linked to:

ALTER [ONLINE | OFFLINE] [IGNORE] TABLE tbl_name
   alter_specification [, alter_specification] ...

alter_specification:
    ...
    ADD [COLUMN] (col_name column_definition,...)
    ...

To find the syntax for column_definition search a bit further down the page:

column_definition clauses use the same syntax for ADD and CHANGE as for CREATE TABLE. See Section 12.1.17, “CREATE TABLE Syntax”.

And from the linked page:

column_definition:  
   data_type [NOT NULL | NULL] [DEFAULT default_value]
   [AUTO_INCREMENT] [UNIQUE [KEY] | [PRIMARY] KEY]  
   [COMMENT 'string']  
   [COLUMN_FORMAT {FIXED|DYNAMIC|DEFAULT}]  
   [STORAGE {DISK|MEMORY|DEFAULT}]  
   [reference_definition]  

Notice the word DEFAULT there.

Like this?

ALTER TABLE `tablename` ADD `new_col_name` INT NOT NULL DEFAULT 0;

table users (user_id int unsigned PK, username varchar(32))

alter table users add column verified tinyint unsigned default 0

Simply add default 0 at the end of your ALTER TABLE <table> ADD COLUMN <column> <type> statement