Here's what I can gather from looking at example sentences from Pleco and Ichacha. I'll only be considering the use of these words as verbs, not as nouns. See the headings for a tl;dr.
表达: to convey
表达: to convey or voice (a feeling or one's thoughts), such as wishes, aspirations, or excitement. This is generally followed by an object, not a clause. Compared to 表示, there's more of a focus of getting your message across.
表示: to show a "social" feeling, to indicate
表示: to show (a feeling), such as friendliness, affection, concern, remorse, or agreement. The emphasis here seems to be more on the outward expression, and seems to be more about feelings that are just socially expected.
This can also mean "to indicate":
发烧表示有病。A high fever indicates sickness.
It can also mean showing non-feelings:
图中表示出顶部的斜度。The slopes of the roofs are shown in the figure.
Unlike 表达 and 表现, which apparently can only be followed by an object, 表示 can be followed by a clause:
她未表示打算做何回应。She has not indicated how she proposes to react.
表现: to demonstrate a quality, to present oneself
表现(出): to demonstrate or display (a quality), such as interest, a sensible attitude, the enthusiasm of youth, or a mistaken belief. 表现 can mean "to present oneself + adv.", "to perform + adv.", or "to behave oneself (well/poorly)", and so it's often followed with a 得. Some example sentences:
他在公司里表现得卑鄙专横。 At the office he was mean and overbearing.
难度表现在许多方面。 Difficulties arise in a number of ways.
她想表现人生的“亲切感”。 She hoped to convey the "immediacy" of existence.
显示: to demonstrate a quality, to display
显示(出): to demonstrate or display (a quality), such as strength, intelligence, beauty, a finite lifespan, many bacteria, or one's heritage.
It can also mean displaying something more literal, such as a speedometer showing 95 mph:
速度计显示95英里每小时。The speedometer showed a speed of 95 mph.
Note that 显示 can be followed by a clause:
他脸上显示出他对演讲感到满意。His face told that he was satisfied with the speech.
调查显示75%的人赞成新法规。Surveys show that 75% of people approve of the new law.