The short answer is there is no magic date when the most of a particular style or type of house has the highest number of inventory on the market. Overall, the number of listings can pick up during the spring market, but along with that typically the number of houses sold increases as well. When looking for a new home, try to think of the inventory and market as more fluid - constantly evolving. If you're trying to see the maximum number of listings in the minimum amount of time, there are two things you can do. First, sign up with a buyer's agent who you're comfortable with and can watch the MLS closely for your potential matches. Second, be flexible in your criteria. Being open to even slightly different criteria can make a huge impact on the number of potential matches for you.
For example, I just checked the town of Manlius for existing single family residential homes with exactly 4 bedrooms, exactly 2 full baths and exactly 1 half bath and came up with 60 matches (with no limits on price). However, if I change the "exact" matches to "minimum number of" 4 bedrooms, 2 full baths and 1 half bath, I get 127 matches! Removing the number of half baths brings it up more, to 149 matches. I use this example without a price because that is usually a buyer's number one criteria that has to be met, and can limit be very limiting by itself.
Please don't hesitate to contact me if you should have any further questions, and good luck in your search for a new home!