Which is the most frequently used tone?
I did some searching for tone distribution chinese and found this post ("What is the distribution of tones in Mandarin Chinese?") on Quora.

One person who responded took a list of characters and extracted, with some programming, the tone(s) for each character. (Multiple pronunciations are permitted and counted separately.) The result he got was:


First tone:


24.2% (counting by characters)
26.9% (counting by appearance in words)

Second tone:


28.1% (counting by characters)
24.7% (counting by appearance in words)

Third tone:


16.3% (counting by characters)
18.3% (counting by appearance in words)

Fourth tone:


31.4% (counting by characters)
30.1% (counting by appearance in words)



However, I am not sure what exactly was counted as a "word".

More importantly, there is also the presence of a tone sandhi. This allows for a "neutral" tone (e.g. in 妈妈) and in a sequence of two repeated third tones, for the first third tone to shift to a second tone (e.g. in 你好). (There are also some more examples with the behaviour of specific characters.) As a result, working off of dictionary entries may not reflect actual frequencies of tones, and moreover, the specific tone used will depend on context.

In any case, this and my experiences with Chinese indicate that there's no one tone that comes up overwhelmingly frequently. However, there are some pointers in this question for tone memorisation, and I particularly like the idea here.