You're asking for a single, large event to cause the shift, which makes sense as the end game. However, I think it's probably more important to establish a series of events and circumstances which set up the situation such that it will culminate with corporate takeover.
Globalization is the word of the day. Large corporations have become international entities with no real ties to their country of origin. They are interested only in the bottom line, and scoff at having any sort of civic responsibility, or really supporting human rights (except when the cameras are trained on them - look at present day Apple factories in China).
Look at the US, for an example of "rabid capitalism": corporations are legally considered "individuals", which allows them to lobby government. They blatantly manipulate politics, and influence major decisions in many areas. This is true all around the world in some way, shape, or form (by having politicians, or their families be involved with said corporations, and thus act in their favor, etc), however in most places corporations don't legally have that kind of power. Thus, it's unrealistic to suggest that corporations would, in one fell swoop, take over the world - or even just the Western countries. People would resist, the military may even rebel. What you need is a good combination of legislation and circumstances.
The Set Up
There's several aspects which need to be staged:
- An economic/security crisis which make people desperate
- Increased corporate rights and freedoms
- Government ineptitude in handling the above crisis
These conditions will be met slightly differently in the US and Europe.
To address the first condition, simply extrapolate current European events: the immigrant crisis of Europe takes a heavy toll on socially generous countries such as Germany, France, and other Northern European nations. Terrorist incidents shake the public's trust in the judgement of their governments (they let the immigrants trample across borders unchecked), as well as their ability to keep them safe (a lot of arrests are made, but dozens of terrorist attacks still occur). This situation sparks a wave of racism and Islamophobia which sweeps many European nations, and which results in many demonstrations and acts of violence against Muslims, and their places of worship - this in turn fuels even more violent acts on the part of their extremists.
As a result of the unstable security situation, the EU's economy suffers greatly. In an effort to bring the economy back on track, and under heavy pressure from the private sector, the EU grants corporations rights similar to those of US based entities. There are groups which protest this, but the average citizen is desperate for a job, and to feed his/her family: they just don't care anymore.
The economic downturn brings a lot of people to desperation. Robberies and violent crime rates rise, as does the aggression with which police deals with the heavily armed (Second Amendment, yay!) perpetrators. Racial tensions rise, with each police shooting, and violent demonstrations and riots take an even greater toll on the US economy. Public trust in the government's ability to keep them safe, and run the country, drops.
Large global corporations band together and announce a US-EU public service project on a massive scale. They begin opening large community centers in the hardest hit European and US cities, and publicly pour a large amount of money (but only a fraction of what they have, not to mention also receiving government funding) into making these a success. They offer employment services, open food banks, child care services, etc. - all while providing jobs for the locals.
To protect these community centers, they set up safe-zones which are protected by private corporate security. They - very publicly - succeed:
- Europe: in stopping several planned terrorist attacks against these centers
- The US: in stopping riots from sweeping over their safe-zones, which keep thousands of women and children safe as rioters rip the city apart around them
Public opinion on the abilities of corporations to keep the peace skyrockets.
As Draco has already stated, corporations already have access to an unprecedented level of information, as well as basically owning the Western media. By carefully manipulating the media, and going so far as to manipulate/edit popular internet sites and forums etc., corporations build a truly impeccable image for themselves, with any incidents involving corporate security violence being swept under the rug, or being very well handled by their PR departments.
All around the US and Europe municipalities ask the corporations to expand their security zones to include schools, hospitals, and other civic buildings. The corporations agree, but request a greater degree of freedom in managing those areas.
The Grand Finale
As stability returns, so does prosperity. Municipal governments already work very closely with corporations, while at a higher level (US federal, and EU government levels) lobby groups are keeping just the right people funded and in power.
After yet another major government fuck up (many troops dying in the Middle East, etc.), which corporations are careful is played up in just the right way in the media, a seemingly "grassroots" movement is sparked:
"The people", clearly seeing that corporations can better manage their countries than their failed, war-mongering governments, demand that these same corporations which have restored their cities take over greater government functions, including foreign policy.
This whole thing is carefully manipulated via social media by the corporations in question, while the alphabet agencies (CIA, NSA, CSIS, Interpol, etc.) have their leashes yanked by corrupt politicians, or other well placed corporate puppets.
After massive referendums across the US, Canada, and the EU, this massive global corporate conglomerate is elected as the managing entity for the Western world. The corporate leaders become public figures of incredible importance.
Each country's military/police force/spy agencies, while technically not falling under corporate rule, now depends on them for funding. Needless to say, the takeover is fast and merciless, with public opinion being manipulated all along the way.
Note: As Avernium points out, corporations are a bit like cats: they compete, not work well together. You would have to also cover how major corporations would initiate this massive effort. No wand-waving required as far as you keep in mind that what they all really want is money. You can imagine a company such as Amazon buying Walmart, then merging with Microsoft/Google/Facebook/Ebay, and becoming a behemoth of unparalleled influence and financial might. Once they have that much power they can bully any other company into bowing down to them (weapon manufacturers, pharmaceutical companies, etc.). Other companies would offer to join their alliance simply to stay on their good side.