A friend asked for this, so I thought it’d be helpful:

All tables with no primary key:

use INFORMATION_SCHEMA;
select CONCAT(t.table_name,".",t.table_schema) as tbl,
from TABLES AS t LEFT JOIN KEY_COLUMN_USAGE AS c
ON (t.TABLE_NAME=c.TABLE_NAME
 AND c.CONSTRAINT_SCHEMA=t.TABLE_SCHEMA
 AND constraint_name='PRIMARY')
 WHERE t.table_schema!="information_schema"
 AND constraint_name IS NULL;

All tables and their primary keys, if exist:

use INFORMATION_SCHEMA;
select CONCAT(t.table_name,".",t.table_schema) as tbl,
c.column_name,c.constraint_name
from TABLES AS t LEFT JOIN KEY_COLUMN_USAGE AS c
ON (t.TABLE_NAME=c.TABLE_NAME
 AND c.CONSTRAINT_SCHEMA=t.TABLE_SCHEMA
 AND constraint_name='PRIMARY')
 WHERE t.table_schema!="information_schema"
order by constraint_name;