Is it possible to know how many seats on a particular flight have been booked? If not, is it possible to know in which months of a year the flights usually are not fully booked?
As far as I know it's not possible for free, but you can buy a subscription to a tool like KVS that can look up availability on any flight:
The numbers in the "Availability" column represent free seats in each fare class: in this case, Business class (buckets J C D) on LH401 is wide open (9 or more seats), while the same flight codeshared as UA8840 has only 4 seats per bucket free. As Doc points out, these don't map 1:1 to empty seats, but the general trends will be pretty obvious: all 0s is packed, all 9s is empty.
The obvious caveat is that, if you look at a flight far in advance and it's still empty, it doesn't necessarily mean it will still be empty on the day of the flight.
Alternatively, you can use any of a number of travel sites with fare prediction capabilities to figure out when you should travel, since cheap fares tend to correlate fairly directly with the number of free seats. The original, Farecast (bought by Microsoft and rebranded as Bing Travel) is now gone, but Kayak now offers something similar.
There are two different things in aviation business:
You can know if the flight has a seat for you to be booked. You can simply know this by visiting any flight booking site, for example: Kayak.com.
This is to know how many seats are available in a given flight for each class, this information is usually available only from the airline's system and NOT from the GDS. I have an access to a system that shows seat availability for the airlines I work for (Amadeus Altea), it will show me detailed information such as this:
Not only that, it is accurate to the point I can see the overbooked seats!
If I use the same system to check other airline's seat availability, I won't be able to get the same full details as above, I will just get a general view of the classes and a simple availability where it will only shows 9 as the maximum number of available seats even if there are much more available seats, for reasons I do not really remember:
When doing the same search on Kayak.com, I got the same results showing 9 seats available:
Bottom line, you will not be able to know the exact number of available seats unless they are 9 or less available seats, and even that is not accurate as mentioned in Doc's comment, unless you have an access to the specific airline's system.