It depends on the laser. 

- If you have a short pulse dye laser and you pump with a ns excimer, the ASE will be on the ns timescale, you can distinguish using a fast photodiode + scope.

- For a laser with short active medium, the divergence of the ASE is usually significantly higher.

- For an excimer, the ASE can't be suppressed easily since the upper state is unstable, hence very short. If you use it as an amplifier, you can have lower the ASE content using off-axis geometry.

- For an XUV FEL working in SASE mode, the jitter is higher and the spectrum is more ugly than a fully coherent seeded FEL.

Could you be more specific what kind of laser do you have?

Parviz, Gabor gave you a pretty detailed review of the situation. In general, the shorter the pulse (especially  if it is a single pulse mode), the tougher is the problem. In ideal, cw (continuous wave) mode, two parameters make your day: narrow line and stable (almost non-fluctuating) amplitude tell you that you have a coherent light from a GENERATOR (i. e. a device with a feedback, similar to your plain Van der Pol rf oscillator). The oscillator/generator works in  a full-blown nonlinear mode  that cuts off the low-competing side bands, and stabilizes the amplitude. In laser it takes many round trips  of light in the resonator. If a single pulse didn't have time to bounce a few time from mirrors, there was not enough time enough to do that. In some (most?) of X-ray lasers, after the amplified reflected signal comes back to the amplifying plasma medium, that amplifier is gone... (it is a bit similar to the excimer laser).  When posting again, can you compare your pulse length/duration (if single) withe the full round time of light to go between mirrors in your resonator?

In multistage laser oscillator - amplifier systems, ASE is often suppressed using a pockels cell/polariser combination and a fast edge driving the cell to prevent ASE for one stage being amplified by the next.

It is very divergent, also, It may be observed before the threshold.

Extreme answer: A laser is not a light source, but, as the name implies, a light amplifier. Of course, a light amplifier is the most versatile building block to set up a light source. And because it is so versatile, you can set up a very precisely defined light source. Usually, but not necessarily, one will apply the principle of feedback with mirrors to set up a self-starting light oscillator having then already the property of producing directed radiation. With additional elements monochromaticity, coherence, polarization, temporal pulse shape and wavelength can be influenced. The laser output starts with "SE", some spontaneous emission, which is then subsequently amplified during the feedback process. In some way all laser output is "ASE", but the term is mostly used for the part of the radiation which has not passed (enough times) above-mentioned additional items. On the other hand, some SE can be amplified to arbitrarily high amounts of ASE in multistage (dye) lasers without using feedback and anyway providing a precise directed beam, as by e.g. the MODE-X broadband modeless dye laser.

About Rolf comment: I think that the laser (Light Amplification by Stimulated Emission Radiation) is a light source, because,  by definition: Amplifier is a device whose output is the input amplified. If, I consider laser like an optical amplifier, then it input need to be light. Light source is a devices that emits light in it output, generated by an internal process.

I can split the laser in three parts, whose functions are more or less explained: 1- A phase of generation de light by a spontaneous process, in a region of generation (active region), 2- Realimentation of this light by reflexions in a cavity, for example, a cavity formed for two mirror, or two reflexives surfaces. This realimentation selects the modes, like the modes of Fabry Pèrot, in the cavity above, and: 3- Stimulated generation of the modes that amplifies these modes. I hope I'm not wrong.

In my opinion, there is no much difference between ASE and laser. Both the cases are referring to light amplification. The only difference is the amplification targets. Specifically, when a fluorescence is passing through a population inverted system, it can be amplified, Its amplification ratio depends on the population inversion lifetime and the length of gain medium. Because fluorescence is isotropic, the ASE is always bidirectional (the propagation direction and inverse of propagation direction). By the way, the generation of intense backward ASE in air with the help of femtosecond laser filamentation is a very important step to remote sensing.  On the other hand, laser means light amplification of stimulated emission radiation. That is to say, when a coherent seed laser passes through the same population inverted system, it can also be amplified. And the whole properties (the direction, polarization.etc.) of amplified light is the same with that of the seed laser. I hope you can find a way to suppress ASE when you understand their generation mechanisms. 

In my opinion the laser is not only a light amplifier, but too it selects modes, by realimentation, then this is the diference between it and ASE.

The main difference is that the  coherence degree ( temporal and spatial) of laser is much higher than ASE, even though ASE may be much more intense. Furthermore, ASE is not dependent on the resonator geometry while laser beam arises from the cavity design.

ASE is amplification without feedback.

Lasing is amplification + feedback.

Siegman, A. Lasers

https://books.google.es/books?id=1BZVwUZLTkAC&redir_esc=y

Dear Antonio

You are right, as a consequence there is better beam quality and narrower bandwidth and higher degree of coherence!

Dear Parviz Parvin,

As Mr.Gabor Kurdi mentioned that, in order to help you more in detail, it may be useful to share what kind of laser you are using. 

ASE behaves like noise.

I tried including this line in my thesis on ASE:

"Lasing generally requires population inversion, whereby more than half of the sites in a system are in an excited state due to electric or optical doping. When this criteria is met, the probability for a photon to stimulate emission exceeds the probability of it being absorbed. At this point, a seed photon (which for ASE originates within the material) can cause an avalanche-like relaxation to the ground state where many sites emit light in-phase with the seed."

but my advisor said it was inappropriate as it explains lasing rather than ASE. Was he referring merely to the first word in the paragraph? This is a frustrating topic for me. I have gotten in arguments with professors who insisted that lasing and ASE are different. Surely they don't mean fundamentally different, or maybe my understanding of the two effects is more fundamental (and hence more abstract) than these experts.

Some of the replies here have referred to the acronym LASER for defining lasing, while others list defining characteristics (like feedback and a cavity) that are not even implied by the acronym. To a young physicist like myself who is just trying to understand what electrons and photons are doing in the material on picosecond timescales, the insistence of some experts that ASE is not just a special case of "lasing" seems to be a matter of ego assertion more than anything else.

After reading these answers, new question came to my mind. What is the difference between the amplified spontaneous emission, amplified stimulated emission and lasing action?