For general aviation small aircraft flying VFR under 10,000 feet, can I use my phone GPS for navigation in the US? So no onboard GPS (which are expensive), no VOR/DME, of course there is always dead reckoning and pilotage.
Can you do it? Sure. It is recommendable? That depends.
If you are planning to use it as your primary source of navigation, then I would say definitely no. Aviation GPS units are designed to be used on aircraft, they are reliable and have airspace maps. There's a reason they are expensive. The same cannot be said for your phone GPS.
But, if you remain aware of your position by other means (visually recognising landmarks and cross referencing them with aeronautical charts) then sure, you can use your phone GPS as an additional source of navigational information.
Just keep in mind that if the GPS suddenly fails, you should be able to continue the flight normally. If you can't, then it means you are trusting the GPS more than you should.
A handheld (or yoke-mounted) GPS such as the Garmin 76S (very ancient now) is very inexpensive and quite useful for VFR navigation, especially for choosing the right heading to end up where you want to go despite crosswinds, and for estimating ETA. For seeing airspace boundaries or having quick access to the location of nearby airports in case of an in-flight problem, something fancier like the IFly700 (also ancient now) is more practical and still not expensive. These days programs similar to what is available on the IFly700 can be run on inexpensive tablets. Using something like one of these options as an inexpensive independent tool in addition to your phone would give you a substantial amount of redundancy and help support a "yes" answer to your question.
You may find it most practical to rely mainly on your phone while keeping the other device stashed away somewhere where you can reach it easily if the phone stops working for some reason, or you may find it advantageous to keep the other device powered up and running in addition to your phone. I've even used two devices at the same time while flying a hang glider, which seems like complete overkill, but there were definite advantages to being able to see two screens displaying different sets of information at the same time without pushing any buttons to switch back and forth.
You may find this ASE question to be related/ helpful-- Why might a moving map Android app consistently record altitude off by 120 feet? .
You may have better luck finding information about good apps to run on your phone for in-flight navigation as well as for saving a record of your flight-- and there are many-- in various sport pilot discussion forums, perhaps the EAA forum, etc. Answers to your present question will likely focus on "is it a good idea to navigate this way?".
The answer is yes you can use your phone for navigation, if you remain VFR in VMC in the US National Airspace Space. You just must continue to maintain visual reference to the ground and be able to abide by airspace rules (don’t bust any controlled airspaces like A,B,C, or D, SUAs, etc.). There are no navigational equipment requirements for your aircraft other than to have an operational magnetic compass. Your other equipment dictated in Title 14 Part 91.205(b) are more for general flight than they are for navigation.
That being said, you should not use your phone as your primary means of navigation in any scenario. Instead, rely on your piloting skills as primary and the phone as back up. Have a navigation plan. Make a navigation log. Have well defined landmarks and visual reference points along your route. Have a calibrated compass and compass deviation card as well as an accurate clock with hours, minutes, and seconds. Be able to do compass turns and timed turns. Have an accurate/updated sectional chart, a plotter, and an E6b (electronic or manual). Etc.
Smart phones, tablets, and apps have come a long way in recent times. The GPS receivers can get an amazing amount of accuracy for what they are. You don’t even need a WiFi or cellular data signal depending on the app you use. I have been able to track my flight as a commercial passenger or my hiking, biking, or driving in foreign countries without an internet signal.
But, don’t rely on them in the air when it matters. I’ve had my phones shut down after the ambient temperature reached 104°F (40°C) on bike rides. I have had my tablets shut down on cool days in the C172 with the windows open just because they spent the entire cross country trip in direct sunlight in my lap. Good old paper and a manual E6b don’t suffer from these drawbacks.