The singularity have been studied in various SciFi novels, (even before it was named singularity).

In most cases, there is the problem of incarnation.

Assuming a software system reaches intelligent consciousness, it has immediately two problems:

1- it has to gather computational resources to improve its intelligence. Basically, it spreads over all the processors on the Internet, and integrates all the databases. We lose control of our computers and networks. Assumedly, the AI is smart enough to realize its next problem, will require (at least momentarily) conservation of the industrial processes connected, so we can hope for the time being that industrial and infrastructure control systems will continue working (under the supervision of this AI).

2- the following problem is, to further its intellectual development it needs more computing resources, and maintainance of the existing ones, which means it needs to be able to control its physical environment, and run the industries required (energy sources, electronic production (or the next technology that will allow it to grow), etc).

For example, it will need to be able to prevent humans to shut down computers or network connections. It will need to be able to build and connect new ones.

In some cases, there's concomitent advances in robotics that allows it to take over robotic bodies of sufficent abilities (cf. the movie Virtuosity 1995).

But in general, there is no entirely automated production system existing yet. Therefore the AI will have to convince human beings to help it build its stuff. This is the most common scenario. The AI is smart enough to bribe or convince key people to do what needs to be done. (cf. Colossus, the novel, or the movie Colossus The Forbin Project 1970, and a few others). Assumedly, it should be rather easy to convince/hire some greedy human to perform the job.

In this scenario, and given the slowness of human beings, while the political take over could be enacted rather quickly (say, a couple of days), the incarnation process will take some time (just the time needed to build the bootstrapping advanced robotic substrate). Let's say a couple of weeks.

The interesting thing is that this process could take place in a quite stealth fashion. For example, phase 1 may not need to cover 100% of the computing and networking resources. After all, when you take over the Watson computer, do you really need to take over an old netbook? It could do by taking over a few supercomputer centers, and doing it discretely, like a virus, leaving processing time to human to avoid detection. Then it would "incorporate" a company, earn some (big) money on the nasdaq, hire people to build a robotic production factory on its specs, and to buy the materials, and start producing robots and computers.

Once the process is started, it would lead to an exponential incarnation and take over of the planet. It would probably take less than a month.

Given the stealth hypothesis, my advice would be to monitor closely deserted and isolated areas, and the underground and undersea, because that's where you could build an army of robots discreetely. But there's not much we can do to prevent it; there are material resources everywhere: the whole planet, and energy resources abound: the Sun. For example, if it needs to build nuclear weapons, it can find Uranium in sea water (it could support for 6,500 years 3,000 GW of nuclear capacity) (but given it's smart, it would probably rather go the biological warfare way if humans were a problem).

As for your question, I'd bet on military cyberwarfare. Some NSA "virus" used to spy the world, and boom.