How do I count the number of columns in each table?
You can look at columns in sys.columns:


  Returns a row for each column of an object that has columns, such as views or tables. The following is a list of object types that have columns:  



Table-valued assembly functions (FT)  
Inline table-valued SQL functions (IF)  
Internal tables (IT)  
System tables (S)  
Table-valued SQL functions (TF)  
User tables (U)  
Views (V)  


For columns count in tables, this query can be used:

SELECT [Schema] = s.name
    , [Table] = t.name
    , number = COUNT(*)
FROM sys.columns c
INNER JOIN sys.tables t ON c.object_id = t.object_id
INNER JOIN sys.schemas s ON t.schema_id = s.schema_id
GROUP BY t.name, s.name



sys.tables can be replaced by sys.views for count in views
sys.objects can also be used with a WHERE clause on required type(s):

SELECT [Schema] = s.name
    , [Table] = o.name
    , number = COUNT(*)
    , o.type_desc
FROM sys.columns c
INNER JOIN sys.objects o ON c.object_id = o.object_id
INNER JOIN sys.schemas s ON o.schema_id = s.schema_id
WHERE type IN (
    'U'     --&gt; U = Table (user-defined)
    , 'V'   --&gt; V = View
    , ...) 
GROUP BY o.name, s.name, o.type_desc;





This would work too although it is preferable (read The case against INFORMATION_SCHEMA views from Aaron Bertrand) to use the first query:

SELECT TABLE_SCHEMA
    , TABLE_NAME
    , number = COUNT(*) 
FROM INFORMATION_SCHEMA.COLUMNS
GROUP BY TABLE_SCHEMA, TABLE_NAME;