How to know a flight is not full?
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.
Your own example shows how unreliable fare buckets can be - the two flights mentioned are the same flight yet one shows 9 and the other only 4. Even showing 9 does NOT mean there's 9 seats available in business - F6 means there's 6 seats in First, so there could be as few as 3 in business, and the airline is willing to oversell business and move people to first if required.
Sure, fare buckets aren't an exact map of free seats (did I claim they are?), but a flight where availability is all 9s is still going to be far less full than a flight that's zeroed out. And if the airline is willing to give you a first-class seat at a business class price, even better ;)
When checking availability for a flight using an access portal other than the airline's, you will only get the maximum of 9 seats even if there are more than that available. That's kinda universal, I do not know the reasons behind it I totally forgot but it must be related to competition or something like that. Anyway I have seen professional reservation agents do some tricks to know an estimate, something about making a fake reservation for 9 and doing that few times till the system refuse then deleting all, not really sure how.
And IIRC some older CRSes (was it Galileo?) will show a maximum of 4, which might be the case in the `UA/LH` flight in the screenshot.