LTE w/ wifi capability is going to be pretty standard on anything offered nowadays.

As for the "big brother type spying stuff", they're all pretty much going to be logging access to their networks.  And even then, if it's the NSA having your stuff you're worried about, they don't need the carrier to be involved anyway.  They'll just snatch it out of the air.  If it's really a big deal for you, you probably shouldn't be using wireless devices of any sort to begin with.

As for a small smartphone, the smallest I can think of that's still a modern smartphone that you can buy new is an iPhone 5.

As for phones with good reputations, Motorola phones have excellent reputations.  HTC has kind of a poor rep in some circles.

As for carriers, Verizon is the 800-lb gorilla; coverage and service reign supreme nationwide.  Spendy though; you pay for that coverage and service. 

The other members of the "Big Four" are T-mobile, Sprint, and AT&T.  T-mobile tends to have a decent reputation out west.  Sprint and AT&T suck in the west, but have a good rep in the east.

Any other carriers are reselling access on one of the Big Four's networks.  Virgin Mobile, for example, resells Sprint mostly.


-T.

Ok, 3g, 4g and LTE are stupid marketing gimmicks. 3G and 4G refer to the "Generation" of the network with plain old Cell service falling somewhere between 1-2G (See marketing gimmick) LTE is the 4th Generation Technology that has gained the most traction, So most phones are going to have LTE anymore... even the non smartphones.

Luckily for you if streaming audio (and video for that matter) are what you want, then any of the current smartphones will take good care of you. I'm partial to Android, but I love to tinker with it, replace the OS periodically etc. Most people don't, if you already have an offering from Apple, then an idevice isn't such a bad investment... never thought I 'd say that out loud, but they do make decent stuff.

Spying? Everyone.  If they operate here in the US they have to comply with legal requests from the NSA (obviously) Your best bet to try to obfuscate (and oddly enough draw in more scrutiny) is to use a custom android ROM (which potentially brings in a host of other issues) built to route all your traffic over the TOR network. Android is operated by Google, and if you run the OS that comes with your device then Google is trying to advertise to you. Apple does the same thing, but not as well.

Size? Well here things aren't looking in your favor, but there is a silver lining. The trend of phones to get bigger is real, and it isn't going away. However that means last year's model, or the one from 2 years ago is going to be a little smaller, and a lot cheaper. Phones are really good now, call quality, performance in every other category... listen to people whine about phones and it is about the cameras... spoiled children. Ahem... back on topic. You can go back a couple years and buy a new phone that was never sold, or a gently used one (look for a model with a removable battery in this case) and go prepaid.

Sadly the best carriers for coverage abuse that position... AT&T and Verizon (Big Blue and Big Red from here on out) are evil. Evil. Both charge too much, but tend to have the best coverage. Sprint is rather slow, and has good coverage where they have it, and terrible coverage everywhere else. T-Mobile has been gaining points for playing almost fair. My personal favorite is a MVNO (they sell other companies' airtime) called Ting (http://www.ting.com) They let you bring a phone from either Sprint or T-mobile (check their website for coverage specifics) My wife has Ting and loves it. I would if I paid for my own phone.

Useful things with a phone? I have a handyman's calculator on mine that will tell me about leveling, filling volumes, how many 2x4s to buy for the size of house I want to build. I have another to sniff out who is on a network, or one to map out where the wireless access points around me are.

Best of luck.

If you view a smart phone as a tool, not a toy, it can quickly become one of your most-used tools.  Calculator?  Language translator?  Calendar?  Alarm clock?  Unit conversions?  All in one place.  And that is before you factor in that you have the answer to any question you could ask, on the internet search function...



Kaso

I'll add that you can get an iPhone 5 cheap, but the battery sucks. 4S (get one used and activate it) is good for what you want, it's smaller and has a much better battery. I'm rocking the 6 Plus, I love the big screen but it actually is a pretty big phone to lug around.




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