It is not possible to conduct online surveillance without setting up a spy app on the smartphone of the subject and it can never been around various legal issues. So it is not a criteria PIs can get involved with, even though we get many inquiries about this matter.
Also, many of those people who are obsessed with inside the cell phone tend to be a bit paranoid and are strongly concerned about hidden online intrigues, rather than logically suspect the subject based on substantial occurrings.
However, those spy apps are easily available in US. So we will introduce tow of the standards of spy apps for smart phones.
Spy apps can snoop around the location, calling history, email history, instant message (Skype, LINE, Whatsapp, etc) history, browsing history, etc.
In order to activate all the functionalities, you need to root the handset for Android, jailbreak for iOSs. Some of the limited functionalities are available for non rooted/jailbreaked handsets.
But the point is you need to physically get access to the handset in order to install the app.
There are some rumors that it can be installed by remote control, but it is just an urban myth.
You must unlock the handset if it is locked. That is another obstacle to get over till you make it in fixing the app.