The most common and most likely answer to this crossword clue is the 11 letter word SPIDER-NETS
I've seen this crossword clue in Financial Times
S
P
I
D
E
R
-
N
E
T
S
Buddy explains!
The straight part
We know that "Flycatchers – the stationary kind" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case flycatchers – the stationary kind = spider-nets.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.