The market for cell phones and cell phone service is an oligopolistic market, at least in the United States.
The market for cell phones and cell phone service is clearly not a monopoly. There is more than one firm that sells these goods and services. Therefore it cannot be a monopoly. On the opposite end of the spectrum, cell phones are clearly not in perfect competition. Cell phones and cell phone services are not homogeneous. There are many kinds of cell phones and many different cell phone plans. This shows that there is competition other than price competition going on in this market.
This leaves us with monopolistic competition and oligopoly. The main difference between these two is in the number of firms that are in the market. In monopolistic competition, there are many firms in the market. This would be a market like restaurants where there are many competitors in each town. By contrast, in oligopoly there are only a very few large firms that dominate the market. This is clearly the case in the cell phone market as there are only a few major companies that provide phones and plans.
Thus, the market for cell phones and cell phone services is an oligopoly.