Is it illegal to install phone spy software on someone's cell phone?
It is HIGHLY illegal to install spy software that in any way records, tracks, forwards, etc. phone calls or text messages on someone's phone without their permission. In fact, it could potentially be a felony, meaning more than one year in jail.
"Wiretapping" is the interception and/or recording of any telecommunications or any oral communication without the consent of at least one party (i.e., a person participating in) to the conversation. (Some states require the consent of ALL parties to the communication, but all states require the consent of at least one party.)
No "wire" and no "tapping" is required. Software which spies on another's use of a cellphone, such as recording or monitoring conversations, copying and forwarding the text messages or emails they receive, etc. would almost certainly run afoul of the wire-tapping statutes.
Wiretapping is serious business. In every state, it is a felony or the equivalent of a felony (not all states still use the term "felony"--New Jersey, for example, does not), which means it is an offense potentially punishable by more than a year in jail and/or by a very large fine. This point cannot be stressed too highly: capturing another's telecommunications or their oral communications is illegal. Any interception or recoding of phone, text, or in-person oral communications is illegal.
It's why, for example, security cameras in stores or office buildings are video only, not sound. Recording conversations on a security camera would violate the wiretapping statutes. Whatever information you think you may get from placing spy software on someone's phone, it's difficult to imagine that it's worth the prospect of serious jail time.
And jail time (and/or a fine) is only the criminal side of the equation. In addition, you could be sued by the person on whose phone you placed the spy software for invasion of privacy and could have to pay them monetary compensation.
If the software is placed with the consent or knowledge of the phone owner, it would be legal--at least in those states requiring the consent of only one party to a conversation in order to monitor or record it. (In "all party" or "two party" consent states, the person on the other end of the call would have to consent, too.) Of course, if the person knows you have placed the software on his/her phone and agrees to it, it's not really "spy" software then--you're not "spying" on someone who knows they are being watched or listened to--but at least it would be legal. So in one-party consent states, you could place such software on, for example, phones you provide to your employees (assuming you are a manager or business owner who provides phones) so long as you notify the employees of the software and get their consent (make it written consent and keep a copy of the consent filed safely, so you can prove consent later if necessary) to do so. However, it's probably safe to say that getting the consent of another person to place spy software on their phone is a rare or unusual occurrence, so it is safer to simply assume that no, you cannot do this.
If you want to know if your state is one-party or two-party/all-party consent state, you can very quickly and easily locate this information via an online search.