A 1837 watercolour of the Great Dust Heap at Kings Cross in London shows a black mountain range, a miniature model of the Alps, surrounded by urban allotments, slum housing and a smallpox hospital. Horses pull carts loaded with debris up to its peaks over which crows swarm. This garbage dump makes an appearance in Our Mutual Friend and the poet RH Horne published an essay about it – "Dust, or Ugliness Redeemed" – in Dickens's journal Household Words. Horne described the underclass of scavengers and sorters who clambered up the dust heap's slopes, ekeing out a living from the city's waste. Everything was recycled: the fine cinder-dust was used to make bricks or sold as fertiliser, the bones went to the soap boiler, old linens were made into paper, the metals were melted down, even the pelts from dead cats had a price. In 1848 the mountain was moved to make way for the current railway terminus and exported to Russia, where the ash was mixed with clay and used to make the bricks that rebuilt war-ravaged Moscow.

Dirt, the new exhibition at the Wellcome Collection, takes us on a fascinating tour through the grimy sublime to illustrate the ways in which we use filth to construct our worlds, social systems and hierarchies. The anthropologist Mary Douglas famously described dirt as "matter out of place", dependant on context for definition. Inspired by this idea, the curator Kate Forde has chosen to focus on six cities at different moments in time, transporting us from 17th-century Delft to the contemporary slums of New Delhi via 19th-century London and Glasgow, interwar Dresden and New York's Fresh Kills landfill site. Dirt is a category violation, a threat to our idea of order and a reminder of our mortality but, however hard we try, it cannot be ignored. Our waste reflects back an uncomfortable truth about ourselves: "The urban physiology of excretion," wrote the historian Alain Corbin, "constitutes one of the privileged means of access to social mentalities."

Compared to the quagmire that was 17th-century London, Delft was thought to be one of the cleanest cities in the world – its obsessive citizens, as one traveller to the Netherlands commented, were "perfect slaves to cleanliness". In The Embarrassment of Riches Simon Schama writes of the spic-and-span Dutch towns and interiors that "shone from hours of tireless sweeping, scrubbing, scraping, burnishing, mopping, rubbing and washing". This war against dirt was pursued with military rigour, as though guarding against the muddy swampland that might bubble up to reclaim the dyke-bound nation. The cleaning regimen, celebrated in domestic scenes by De Hooch and Vermeer, expressed a deep Calvinist fear of shame. Cleanliness was, as the saying went, next to godliness.

Seventeenth-century Delft was also the home of Antonie van Leeuwenhoek, the draper and scientist who ground his own magnifying lenses to inspect the quality of cloth and discovered by accident a wonderful active world of teeming bacteria that he called "animalculae". Microbiology would initiate a scientific war on dirt. Filth had always been associated with illness but, before the late 19th century when Louis Pasteur proved the germ theory of the transmission of disease, contagion was thought to be borne by noxious air, or miasma. The danger to health was believed to come from the stinking exhalations that accompanied decay, not from the putrid matter itself – and each infectious gas was thought to have its own distinctive stench. Microscopy showed an invisible, pulsating world of dangerous pollutants that spread disease by other means.

William Heath's 1828 cartoon, A Monster Soup, shows a lady looking through a microscope at a sample of drinking water from the Thames. She drops her teacup in disgust when she observes the menagerie of diabolical beasts it contains – hydras, gorgons and chimeras. A decade earlier it had been decided to connect house drains to London's sewers that had previously been used to carry off rainwater. Thanks to the increasingly popular flush toilet, the river was soon turned into a stinky swamp and cholera became rife. The much-feared disease, which caused vomiting and diarrhoea, would literally suck the fluid out of its victims, turning them anaemic and blue within a matter of hours. In 1849 an epidemic killed 15,000 people.

Despite jokes about contaminated drinking water, the miasma theory remained prevalent and people continued to drink it. Medical statisticians tried to prove that cholera was spread by polluted air by showing that you were less likely to die the higher you lived above the Thames, because at lower altitudes the miasma originating in the fetid water could gather and ferment. However, it was acknowledged that little was known about the disease. After another epidemic in 1853, the Lancet medical journal asked: "What is cholera? Is it a fungus, a miasma, an electrical disturbance, a deficiency of ozone, a morbid off-scouring of the intestinal canal? We know nothing. We are at sea in a whirlpool of conjecture."

The following year John Snow, Queen Victoria's anaesthetist and the "father of epidemiology", conducted research that would prove that people became infected when they swallowed faecal water, the bacteria that lived in the excrement multiplying in their intestines with fatal effect. Snow drew a "ghost map" of Soho that recorded the many deaths due to cholera over a month and a half, illustrating how these clustered around the Broadwick Street pump that had been dug, it turned out, next to an old cesspit. When the handle of the pump was broken, the cholera subsided.

In 1858, after privies and cesspools overflowed during a particularly hot summer to cause the "great stink of London", which nearly saw Parliament decamp to Richmond, a new system of sewers was commissioned. Joseph Bazalgette's vast network of tunnels, which used 318m dust-filled bricks, deposited the city's shit into the eastern reaches of the Thames. The last attack of cholera was in 1866 in a part of east London where these works had not been finished. (After flooding in the summer of 2004 flushed 600,000 tons of raw sewage into the Thames, the digging of a new £3.6bn "super sewer" the size of the channel tunnel was begun.)

If bacteriology explained the spread of disease, it created new anxieties. The invisible organisms causing infection and illness were no more apparent or comprehensible to most people than noxious gases. Housewives relied on scientists to help them protect their families and combat bacteria with a proliferating range of new chemical disinfectants. In 1911 more than 5 million people visited the First International Hygiene exhibition in Dresden, where information about anatomy and healthcare was disseminated, and where many cleaning products were on display (it was organised by an industrialist who manufactured them). The poster advertising the exhibition featured as the symbol of hygiene an all-seeing eye.

In 1930 many of these exhibits were rehoused in Dresden's Deutsches Hygiene museum, a series of gleaming white modernist buildings that looked purposely sterile against the backdrop of the old baroque city. When the Nazis seized power three years later the museum was co-opted by the Nazi propaganda machine, keen to assert the importance of heredity, eugenics and "racial hygiene". Disgust took on metaphorical uses; in opposition to the Aryan master race, the Jews were depicted as unclean – germs, cancer cells, rats. In one photograph on display at the Wellcome, Nazi officers and party members look on as Jews, to further identify them with dirt, are made to scrub the pavements of Vienna.

Elsewhere, modernist architecture represented the marriage of science and ambitious social purpose in the war on filth and disease. In a 1943 poster Berthold Lubetkin's Finsbury Health Centre (1938), today sadly neglected but once a model for the NHS, is shown as a brilliant white façade veiling a Victorian world of ruin, disease and neglect. The industrialised west hoped to hide its waste, and everything associated with it, behind a technological front. We throw our rubbish away, but have little idea of where it ends up. In 1948, for example, New York started sending tons of its pungent detritus by barge to a marshy part of Staten Island called Fresh Kills. In 1987, 29,000 tons of rubbish a day was being dumped there and spread by 650 sanitation workers across a sprawling 2,200-acre site. It could be seen from space with the naked eye but few New Yorkers knew it existed.

In 2001 Fresh Kills was decommissioned, but after the 9/11 attacks the rubble, twisted beams and crushed cars from the World Trade Center were dumped there. Unlike London's Great Dust Heap, very little of Fresh Kills's rubbish was ever recycled, but the site itself is to be detoxified and reused, covered with grass to create a park over an enormous time capsule of post-industrial waste. Even so, displaying our deep horror and ambivalence to dirt, many people protested that a garbage dump was an unsuitable plinth for a memorial to heroes.

The Wellcome exhibition makes clear that the water-borne diseases that have been banished in the west are still commonplace in the developing world, where 2.6 billion people have no access to toilets and diarrhoeal diseases kill a child every 15 seconds. In India, for example, despite the practice being outlawed in 1993, there are still a million manual scavengers who use straw baskets to remove night soil from dry latrines. This job is performed mainly by women from the lowest sub-caste of the Dalit, who are rendered doubly untouchable by their association with dirt.

But dirt, as Mary Douglas showed, can also take on symbolic import as a powerful symbol of purification and renewal. Each year, in a sacred rite, statues of the goddess Durga, made from dirt gathered from prostitutes' thresholds, where men are thought to have cast off their virtue, are taken in procession to the Ganges where worshipers look on as they dissolve in the muddy water.

Dirt: The Filthy Reality of Everyday Life is at the Wellcome Collection, Euston Road, London N1 from 24 March to 31 August. www.wellcomecollection.org