A dictionary of Confusable Phrases: More Than 10,000 Idioms and Collocations

catch a look • catch the eye
catch a look —
1
. notice smb.’s glance: I caught a fleeting look
of relief from Sylvia. 2. look at a person or thing: He turned
to catch a look at her sweet face.
catch the eye —attract attention: The thickets of Licuala
catch the eye with their striking foliage.
catch a tiger by the tail • take the bull by the
horns
catch a tiger by the tail —(also: have a tiger by the tail) engage
in an undertaking which may prove extremely dangerous or
difficult: You’re taking on an organization with reserves you
know nothing about. How do you know you won’t be catching
a tiger by the tail?
Note: 
The expression does not correlate in meaning with the
phrase catch an eel by the tail—(also: hold an eel by the tail)
attempt to control an elusive person or situation: He is not easily
managed, but if he suspects you are trying to take him in, you
may as well catch an eel by the tail.
take the bull by the horns —deal with a difficult situation in
a bold and direct manner rather than evade it: He had not, as
the phrase goes, taken the bull by the horns, or advanced in
front of the enemy’s fire.
catch on to someone • catch on with someone
catch on to someone —understand the real nature of a person:
He was a real talent and I hoped that the world would catch
on to him.
catch on with someone —become popular with people: I hope
our new product catches on with children.
catch on to something • catch up to something
catch on to something —figure smth. out; understand smth.:
His sagacious mind immediately recognized and caught on to
the only plan of salvation in sight.
catch up to something —(also: catch up with something ) 
1
.
overtake a bus, etc. by walking fast: Jane caught up to the bus
that had almost left her behind. 2. keep pace with smth.: Quite
frankly, at our current pace, we will never catch up to fixing
our infrastructure deficit.