special relativity and time dilation

So I'll explain what I think I know and what I am unsure about.
I understand that it is the stationary observer that always records the proper time, and an observer moving at a constant velocity v relative to the event will always record a longer time.
However, I was reading about an experiment to measure muon decay, whereby the time was recorded for the muons to travel from the upper atmosphere to the ground from the perspective of an earth observer, and it showed that, from the frame of reference of the muons, less time must have passed. However, what's to say that the 'proper time' occurs from a stationary observer relative to the mouns, rather than the other way around? I.e., the situation could be viewed in terms of the earth travelling towards stationary muons, in which case it could be said that the earth observer would measure the proper time, therefore meaning that a longer time would have passed for an observer stationary relative to the reference frame of the muons.
To sum this up, it appears that in many situations involving time dilation, it is unclear as to which time will be the proper time. Could anyone shed some light!?!

What do you mean by 'proper time'? Whereas an observer records a particular time for, let's say a rocket moving close to the speed of light, time will actually be travelling more slowly for the person on the rocket. What the 'proper time' is depends on the person, at least that's what I thought the whole point of relativity is, time is relative to the person depending on the situation.

As for muons, they actually play part of the experimental evidence that show that this relativity and time-dilation actually occurs. Muons have short life times and decay very quickly, so much so that they do not have time to reach ground level from the Earth's atmosphere. Therefore without special relativity, muons should never be detected on the ground. However this is not the case as muons have been detected. The reason being that, because the muons are travelling close to the speed of light, for the muons, they are travelling slower and so decay more slowly. So in effect, muons have a longer half-life due to their high velocities.

Although I have not (yet) studied relativity, I have simply replied as someone who loves the topic but I look forward to other responses.

Also this clip might be of some interest (if you haven't seen it already)

http://www.youtube.com/watch?v=jlJNsRZ4WxI

Thanks for the response.
By 'proper time' I mean either a)the time recorded by an observer stationary relative to the event, or b)the shortest recorded time for an event to take place.

In some situations I have no problem identifying which will be the proper time. For instance, when a person is on a high speed train and switches on a torch for exactly 2 seconds, the time recorded by the person will be the proper time as there is no relative movement between the person and the torch radiating light, and any other measured time (for example, by someone on the train platform and recording the same event), will be greater.

However, in the case with the muon decay, I think it could be argued either way as to whom is the stationary observer relative to the event is, and therefore debatable as to who measures the least time, or proper time (similar to the twins paradox in the video you linked). For that reason, it seems equally plausible that, from the reference frame of the muons, the time passed for the muons to travel a set distance could be more, because time will appear to be running slower for an earth observer.

This is problematic because in these sorts of exam questions, I can never tell whether to multiply or divide by the relativity factor.
Thank you for your time!