SWT had the largest percentage of outright complaints from the tweets from its customers according to a survey of 92,000 tweets by social media monitoring company Brandwatch.
SWT, which operates trains from London's Waterloo station to Surrey, Hampshire and Dorset, attracted 44% of all the tweets studied, with 36% of the SWT tweets being outright complaints.
The next least popular company, according to tweets, was First Great Western, followed by Southeastern.
The company with the most positive number of tweets was East Coast.
Overall, 29,000 tweets were negative, with only 5% being positive.
The top five complaints were late-running trains, no air conditioning, broken down trains, being late for work and the rush-hour crush.
Joel Windels, lead community manager for Brandwatch said: "It's no surprise that people love to moan about public transport in Britain. It's up there with the weather as one of the things we like to grumble about most.
A spokesperson for the South West Trains-Network Rail Alliance said: "We run the busiest franchise in the UK and one of the busiest train companies in Europe, carrying almost 210 million passengers a year. The volume and amount of communication from our customers reflects that.
"We are already responding to many of the issues raised by our passengers - such as overcrowding, disruption and information provision - with hundreds of millions of pounds of investment in extra train carriages, infrastructure improvements and customer communications.
"Passenger satisfaction remains at a high level and figures for 2012-13 published by the Office of Rail Regulation show that SWT has the fourth lowest complaint rate of the 19 franchised train companies in the UK.
"It is impossible to compare the performance of companies which have a largely long-distance leisure customer base with those which have a high proportion of commuters."