The most accurate way to do it would be with a pitot tube to measure wind velocity and then to contrast that against speed-over-ground from your GPS or wheel speed sensors.
Before one laughs about the science-fiction nature of this, there's a product that does so: The PowerPod (link is to DC RainMaker's review of it).
The Isaac software that comes with the PowerPod allows you to extract effective wind speed and direction amongst other things.
bicycles.stackexchange.com/users/11160/RoboKaren
We like tailwinds as the effort to ride at a given speed is lowered.
In the same token we like slight downward slopes.
If you base your riding on effort, rather than a fixed speed, you will automatically utilise a slight tailwind.
If you however ride at a fixed speed, you simply had an effortless ride.
In other words, you missed nothing.
bicycles.stackexchange.com/users/30402/gschenk
@Daniel has made several comments that capture what experienced cyclists have learned.
You think something like
I'm going well , or killing it , or even (as happened today)
this looks like it's uphill , while blasting along.
When you stop you discover the awful truth.
It's a headwind home .
The reality is that we nearly always have a headwind , especially as Daniel says, when climbing a hill, because we're cyclists.
And we usually have a headwind going there, and coming back.
That one in a hundred tailwind case is not the first thing we think of.
Yes, vegetation can tell us.
But in many cases in a built up environment we can't see any vegetation.
But it wouldn't be the first time I've seen the vegetation looking like I have a tailwind when I actually have a headwind.
Because I'm a cyclist.
bicycles.stackexchange.com/users/8273/andy256
If you are biking along and you encounter a terrible smell -- look around and realize that it's you yourself that you're smelling -- then you're in a tailwind -- and that it's time you washed your bike jersey.
Thank you for the downvotes, I'll show myself the door. :-)
bicycles.stackexchange.com/users/11160/RoboKaren
You can check wind direction from bending of vegetation, it stays bent to the wind for longer periods than it springs backwards in eddies, attach a white cotton/wool thread to the handlebars and check it's movement every time the bike slows, feel the wind in left ear, right ear, forwards and back, and check a weather graph with wind direction and speed expected for every hour of the day, and go out when it's max tailwind and come back when it's calmed on the return.
mostly the wind is as likely to be against you as with you, although checking the daily forecast can mean taking advantage of massive tailwinds to get some place very fast and then coming back in relative calm.
If you travel through an entire country you can get a map of the prevailing winds for every season and for every region.
bicycles.stackexchange.com/users/31261/aliential
The simplest way is simply stop in an open space and feel where the wind is coming from, using your face.
Downside is that any trees or buildings will make the wind veer, and traffic makes its own wind.
Plus you have to stop (anathema!)
Flags work really well, because they're often up high and in the real airstream.
You can also check the local weather report for your area.
For me that's http://www.metservice.com/towns-cities/christchurch/christchurch
but you should have something similar for your location.
bicycles.stackexchange.com/users/19705/Criggie
Fresh answer - I did a bit of a ride yesterday
and it was quite warm.
So I ended up rolling my pants cuffs up to my knees for added cooling.
An unexpected side effect is that on my shins I could feel air gusts and eddies that were different
depending which way I was riding.
Its really hard to explain, but in a crosswind I could feel a wind vortex on the inside of my left leg and on the outside of my right leg.
In a headwind I could feel the wind curling around behind my shins, and in a tailwind there was no particular sensation either way.
So if like me you habitually wear pants, try some shorts, or roll up your pants cuffs and see how it feels.
Mind out for sunburn though
- that's a whole separate painful issue right now.
bicycles.stackexchange.com/users/19705/Criggie
I think I missed a lot of opportunities to go faster when cruising and not taking advantage of tailwind. I find that if I put short effort in acceleration, the tailwind will take care most of the speed maintaining work.  For big tailwind I can usually see leaves rolling on along my general direction. Lately, I think if wind noise is more quite, I might be in tailwind?  So how can you tell if you've got tailwind or wind direction in general? (without flag anywhere nearby)
How to know you have tailwind?
bicycles.stackexchange.com
<wind>
bicycles.stackexchange.com/questions/44967
bicycles.stackexchange.com/users/3042/imel96
S0: You can check wind direction from bending of vegetation, it stays bent to the wind for longer periods than it springs backwards in eddies, attach a white cotton/wool thread to the handlebars and check it's movement every time the bike slows, feel the wind in left ear, right ear, forwards and back, and check a weather graph with wind direction and speed expected for every hour of the day, and go out when it's max tailwind and come back when it's calmed on the return. The most accurate way to do it would be with a pitot tube to measure wind velocity and then to contrast that against speed-over-ground from your GPS or wheel speed sensors.  If you travel through an entire country you can get a map of the prevailing winds for every season and for every region. The daily weather forecast will tell you about wind direction. You could use a device to measure wind speed or check the local weather reports showing wind sped and direction, then take advantage of the expected tailwinds when planning your route. Flags, vegetation and the feeling of the wind on your body can be a good indication of wind speed and direction.
You could use a device to measure wind speed or check the local weather reports showing wind sped and direction, then take advantage of the expected tailwinds when planning your route. Flags, vegetation and the feeling of the wind on your body can be a good indication of wind speed and direction.