From Wikipedia

quote

In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force. The effects of this force are easily observable at both the microscopic and macroscopic level, because the photon has no rest mass

unquote

Saying that photon does not have rest mass does not mean that you can catch the light
It must be understood that photons are the only entities having only the mass due to its kinetic energy
Material particles, such as electrons ,have both rest and dynamic mass

in classic mechanics momentum = mass x velocity thus the claim photon does not have mass but it has a momentum would not simply make any sense

The definition of momentum and mass have however changed to keep up with the new experimental evidences
in relativity momemtum is

P = k x Mo x V

Where k = 1/sqrt(1-(v/c)^2) and Mo is the rest mass
if a photon is also a particle then Mo = 0 since v = c
While p = 0/0, the equation falls into a singularity
The momentum of the photon can not be therefore calculated

Einstein to explain the photoelectric effecte assumed following Planck , that

E = hf

Where E is the kinetic energy of the photon
, from which you can get P and M

Besides, in his initial formulation of the equation, Eistein wrote

Mo = L / C^2

Where Mo is the rest mass and L the kinetic energy
Einstein did not use the letter E even though it was also popular in Germany at the time ,to indicate a generic form of energy
Letter L stands for LebenKraft which specifically mean kinetic energy i.e a mass related form of energy

if a body absorbs ( emits ) a radiation of energy L then the body absorbs (loses) the mass L / c^2
Thus photons in motion have mass
Even though Einstein never said that an eletromagnetic radiation is "massive" this was implicit in his words

Summing upr the characteristics of a photon were

rest mass Mo = 0
energy E = hf
mass M = hf/c^2
momentum P = hf/c

If highly energetic photons collide , their direction change same as they were material particles
Diffusion of light was definitley excluded by classic Maxwell's equation

Ray of light are deviated by the gravitational field of the sun same as material particles

The scenario changed again with modern quantum physics
Actually some other experimental evidences are not consistent with the assumption of photon havin a mass in motion even though the "dynamic" mass related properties of the phorons can not be ignored thus:

virtul particles popped up and they have... mass