'Evade' contains an element of sneakiness, for me.
Tax evasion is illegal, as it is done by breaking tax laws; tax avoidance is not. (It may be unethical, but it is not illegal).
'Refrain from' means 'avoid doing, even if you really want to'. (It literally means 'hold yourself back').
'Please refrain from touching the exhibits' would be common in a tactile sculpture exhibition.
'Abstain from' contains an element of avoidance of vice or sin, and usually has religious or ascetic overtones.
Members of the Total Abstinence movement in Ireland abstained from all forms of alcohol/Mormons abstain from caffeine.
It is usually a solemn verb, and indicates the signing of a pledge, or a profoundly held religious belief. It is more often used with a noun, although it can also be used in the gerund form.