AT&T uses LTE bands 2 (1900), 4 (1700), 5 (850), 12/17 (700 a, b) 29 (700 d) and 30 (2300).
Bands 2, 4 and 12 are AT&T's primary LTE bands. Band 17 is the subset of band 12 that AT&T uses. AT&T originally identified its 700 Mhz spectrum as band 17 but now identifies it as both 12 and 17 which allows phones that support either band 12 or band 17 to use it.
Bands 5, 29 and 30 are used in areas where AT&T doesn't own any band 2, 4 or 12 spectrum and to increase capacity and speeds.
AT&T's 3G and non-LTE 4G (UMTS, HSPA and HSPA+) services operate on the 850 and 1900 Mhz bands. AT&T shut down its 2G (GSM, GPRS) service at the begining of 2017.
T-Mobile uses the 1900 Mhz band for GSM (voice) and EDGE (2G) and UMTS (3G) data. It uses the 1700 (AWS) band for UMTS (3G) data rather than 1900 Mhz in some markets.
For LTE, T-Mobile uses bands 2 (1900), 4 (1700), 12 (700 a,b,c), 66 (1700 Mhz) and 71. T-Mobile's primary LTE band is band 4. Band 2 is deployed in former 2G only non-urban areas where T-Mobile doesn't have any band 4 spectrum. In some, maybe all, of these band 2 only LTE markets, 3G is not offered, only 2G and LTE.
In some areas Band 12 is T-Mobiles only band of any kind. To make calls in these areas a phone with VoLTE support is required. There's a map of T-Mobile's band 12 spectrum here. Band 12 is a lower frequency band that provides extended range and improved building penetration compared with bands 4 and 2.
Band 71 (600 Mhz) is T-Mobile's newest band. Band 71 deployment started in the third quarter of 2017. T-Mobile says it will deploy band 71 nationwide, including areas where T-Mobile currently has no coverage. It will take several years to complete band 71 deployment. As of September, 2017, the T-Mobile branded version of the LG V30 is the only phone that supports band 71.
As for Sprint and Verizon, you can pretty much forget about using phones from other operators on either of their networks. With a few exceptions (some Motorola phones, iPhones from the Apple Store and the Nexus and Pixel phones from the Play Store), both operators will only activate phones that they have sold. It's still useful to know which bands Sprint and Verizon use when shopping for phones as older phones nay not support all of the bands currently used by either operator.
Verizon uses 850 Mhz and 1900 Mhz for voice, 1xRTT (2G) and EVDO (3G) data
For LTE, Verizon uses bands 4 (1700c), 5 (850) and 13 (700). Band 13 is Verizon's primary LTE band. Band 4, which Verizon markets as XLTE and band 5, are used to provide increased speed and capacity in urban areas.
Sprint uses the 800 Mhz and 1900 Mhz bands for voice, 1xRTT (2G) and EVDO (3G) data
For LTE Sprint uses bands 25 (1900), 26 (800) and 41 (2500). Band 25 is Sprint's primary LTE band. Bands 26 is used to improve building penetration.
To be sure that a phone will work to its fullest or at all with a given carrier you need to know which bands it supports.
For phones made for the US market PhoneScoop.com is my favorite resource for checking specs. It provides a full page of detailed specifications for virtually every US phone model. Supported bands are listed in the Modes section of the specs. Click the + more detail link to see which LTE band numbers are supported. With LTE It's not enough to refer to bands by their approximate frequency (850, 1700, 1900), as multiple incompatible LTE bands can exist in the same frequency ranges. LTE bands can only be positively identified by their LTE band number.
For checking band support for non-US market phones I like PDAdb.net and frequencycheck.com Be sure you are looking at specs for the phone's exact model number rather than its marketing name (Samsung Galaxy i9500 rather than Samsung Galaxy S4), as most phones come in multiple versions supporting different bands.
"Universal" Unlocked phones:
When comparing phones you will quickly discover there are relatively few truly universal unlocked phones that are sold unlocked and work across all operators. Work is a relative term as none of these support all the bands used by all the carriers. Here are the ones I'm aware of that are whitelisted by Sprint and Verizon and support at least the major bands used by all four national carriers:
- Apple iPhone 6s (A1633) and 6s Plus (A1634) lacks T-Mobile bands 66 and 71
- Apple iPhone SE (A1622) lacks Sprint band 41, AT&T band 30 and T-Mobile bands 66 and 71
- Apple iPhone 7 (A1660), 7 Plus (A1661) lack T-Mobile bands 66 and 71
- Apple iPhone 8 (A1863), 8 Plus (A1864) and X (A1865) lack T-Mobile band 71
- Alcatel Idol 5s (6060S) lacks AT&T lacks bands 29 and 30 and T-Mobile band 71
- BlackBerry KeyOne (BBB100-3) lacks EVDO (Sprint, Verizon 3G) and T-Mobile bands 66 and 71
- Google Pixel (G-2PW4100) and Pixel XL (G-2PW2100) lack T-Mobile bands 66 and 71
- Google Pixel 2 (G1011A) and Pixel 2 XL (G1011C) lack T-Mobile band 71
- Google Nexus 5X (H790), lacks AT&T band 30 and T-Mobile bands 66 and 71
- Google Nexus 6 (XT1103) lacks AT&T band 30 and T-Mobile bands 66 and 71
- Google Nexus 6P (H1511), lacks T-Mobile bands 66 and 71
- Motorola Moto X Pure Edition (2105) (XT1575), Moto G4 (XT1625) Moto G4 Plus (XT1644) and Moto G4 Play (XT1607), lack AT&T bands 29 and 30 and T-Mobile bands 66 and 71
- Motorola Moto E4 (XT1767), Moto E4 Plus (XT1775), Moto G5 Plus (XT1687), Motorola G5S Plus (XT1806) and Moto X4 (XT1900-1) lack AT&T bands 29 and 30 and T-Mobile band 71
- Motorola Z2 Play (XT1710-2) lacks T-Mobile band 71
- Orbic Wonder (RC555L) lacks AT&T bands 29 and 30 and T-Mobile LTE bands 66 and 71
- Samsung Galaxy Note 8 Unlocked SM-N950U lacks T-Mobile band 71
- Samsung Galaxy S7 Special Edition (SM-G930U) and S7 Edge Special Edition (SM-G935U) lacks T-Mobile bands 66 and 71
- Samsung Galaxy S8 Special Edition (SM-G950U) and S8 Edge Special Edition (SM-G955U) lacks T-Mobile bands 71
- Essential Phone PH-1 lacks T-Mobile band 71
Sources: Cellular Binder, TmoNews, Apple, Wikipedia, FrequencyCheck, Howard Forums
Image: Cell phone tower in Oregon by M.O. Stevens some rights reserved CC-BY-3.0 via Wikimedia Commons