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I seem to recall learning that it is permissible to use a cell phone as a watch on Shabbos as long as the time is always displayed on the front and no buttons would need to be pressed in order to observe the time. Of course, in order to carry the phone, there would need to be an eruv. The argument was that the phone is not muktzeh if it is being used specifically for the purpose of time-telling.
Has anybody else ever heard this, and is this idea sourced anywhere?
I would like to add that the mishna brura in 'סימן שח' סעיף קטן יב says that one should not use an item that is a kli shemelachto lissur if a kli that is muttar is available. So this question can only be relevant if one doesn't have a watch.
So this is actually several different questions. In order from more to fewer poskim would say "yes":
1) Is it permissible to look at a cell phone that happens to be displaying the time on Shabbos?
2) Is it permissible to pick up your cell phone within your house and move it around for the purpose of having the time displayed elsewhere?
3) Is it permissible to press a button on your cell phone to cause the time to appear?
4) Is it permissible to carry your cell phone (where there's an eruv) and look at the time on it where other people can see you?
The answer to the first question is for sure "yes." Questions 2-3 depend on why you think using electricity on Shabbos is forbidden. Most cell phones today, especially so-called "smartphones," have gyroscope sensors in them and you will be causing circuits to change state merely by jostling them around.
As for question 4, it seems to me that if your cell phone is displaying the time, then it is also on, and so, if a non-Jew calls you, then it will ring, and that would be zilzul shabbos for sure, all other issues aside. Many phones have "airplane mode," which complicates the factors even more.
These kinds of issues are so complicated that only ravs who are also electrical engineers are really qualified to poskin on them, and how many of those are there? Also, each phone would have to be individually certified for "clock use on Shabbos" since the factors are sufficiently complicated that a general ruling would not cut it. In this case, I cannot see how one could justify a "yes" answer to number four under ordinary circumstances.
