I think that Google having most of the search market, along with them having one of the dominant phone platforms could potentially wind up being looked at.
Much of the investigation into Microsoft was that they bundled and integrated internet explorer into their operating system, which leveraged their OS advantage to boost their browser market.
Since Google services are completely integrated and default in the Android operating system, it might be something they would look at. Though in this case they are more leveraging the search advantage to boost the phone market share. And they use their search engine home page to market all of their tied in apps.
Actually, Google services are not necessarily integrated or the default. In fact, the largest maker of Android phones, Samsung, has their own services -- such as their browser, phone app, texting app, etc. as the default on their phones. While I do believe Google asks manufacturers that use Android to include the Google apps (services), they are not required to, and some manufacturers do not, make them the default. Additionally, almost 45% of smartphones in the US are made by Apple, which ship with no Google apps installed on them. Between Apple and Samsung, that makes nearly three quarters of US smartphones that do not use Google apps by default.
What percentage of the phone market did AT&T command?
Prior to the AT&T breakup, they owned and operated 100% of the US local telephone network (and Canada's). You can argue that the network was broken up between several AT&T subsidiaries -- but the fact remains they were owned by AT&T. In fact, it was in denying connections to MCI who was attempting to compete with AT&T in data transmission that led to the breakup of AT&T. So they not only had a clear monopoly but were caught trying to stifle competition.
What percentage of internet searching and advertising does Google command?
What percentage of internet videos does Google control?
As for Internet searches, 88% of US searches are done on Google. I think the big issue with trying to call that a monopoly is, as was stated above by LoAmmi, they have that number because people choose to use them -- not because there aren't other choices.
YouTube is harder to say -- largely because there are so many other sources to watch video online. What makes YouTube unique, and why people want to call them a monopoly, is because of their success in having people upload video content to them, and then to allow people to watch those videos for free. I don't think you can really call YouTube a monopoly, though, since there are so many other sources to watch videos, and a large number of people watch a huge number of hours of videos from places like Netflix, Hulu, Amazon, etc -- not to mention all the OTA and cable channels that offer content streamed over the Internet.
If Google (a really does have a monopoly, it likely is in their digital advertising company -- which provides the revenue for Google search and YouTube. While that may be a monopoly and likely should be investigated, I understand it is not politically "sexy" -- since that isn't where politicians can claim bias. I'm also not sure that splitting them off would make fewer people use Google Search or YouTube (it might hurt YouTube's payouts -- though I'm not sure if that would increase competition or just reduce the amount of videos uploaded, since there would be less money for creators).
Last, there is an "issue" with the Internet in that services become so large precisely because they get a large number of users. People use Facebook because it is the one place all their friends and families belong to; but prior to Facebook you had MySpace -- and if a better social media site comes along and can get a lot of users, they'll likely make Facebook the "new" MySpace (as in obsolete).
This also shows a real problem -- how do you split something like Facebook up. Chances are, whichever of the new "baby Facebooks" had the best content would cause everyone to join that one -- and you'd just have a new Facebook (with the other "baby Facebooks" dying out). While YouTube might be easier to split (since you could split perhaps on subject matter -- YouTube Music, YouTube Tech, YouTube Politics, etc -- that still likely wouldn't help competitors to break into the market. The issue is, when we are talking about social sites, people will want to be where all their friends and family are -- which leads to everyone using the same sites.