Lipo powered earphone amp
Let's consider the headphones.


Earbuds. No need for a dedicated amp since earbuds audio quality is terrible anyway.
In-ear rubber-nipple thingies. I picked a random one off the net: Sennheiser CX300 is 18 ohms and delivers a whooping 118dB for 1V RMS.
Real headphones. OK, let's pick some not too expensive ones... HD4.20 again 18 ohms and 118dB for 1V RMS.
Some cheaper ones HD2.10 26 ohms and 110dB/1VRMS


Sooo... if you use the latter ones, listen to an average level of 80dB which is already loud, with peaks around 100dB (if your music is not too compressed), then:


RMS voltage will be -30dBV, ie 31mV RMS
Peak voltage around -10...-3 dBV (like half a volt)
Peak current won't be huge either, like 30mA.


Now, let's look for an opamp.

The "maximum output current" specification is useless here. What we want is an opamp that can output the required current with the least amount of performance degradation. An opamp picked at random may have good distortion figures, but when driving relatively high currents (for an opamp) with low headroom, it's an entirely different story.

For example, TL072 isn't so bad with a 10kOhm load, but if asked to drive any kind of actual current, its performance simply ceases to exist.


  The 32 ohm speaker at 3.7V would have about 100mA peak current, which is too much for a headphone. 


You will go deaf before the voice coil burns.

The resistor which everyone puts at the output of headphone amps actually degrades performance, but it has a rather important purpose: protect the output stage when the output is shorted.

...and every time you insert a 3.5mm jack into the connector, the output will short during insertion. This is a "feature" of this connector, you gotta live with it. If the connector is mis-inserted, the short can be permanent. You must select an opamp that will survive this, so you can get rid of the resistor for better sound.

(Explanation: headphone impedance varies with frequency like every loudspeaker. With a resistor in series you got a voltage divider. Voltage on headphone thus depends on its impedance at the frequency of interest. So you get bumps and dips in the frequency response).

You mention a discrete amp. With such a low supply voltage, discrete design is very complicated. An opamp would offer much better performance.

For example,

OPA1688 is a nice opamp with high output current and rail to rail output; it is specified to drive headphones. Beware the input is not really rail to rail, it should not go above Vcc-2V, but this should be OK.

If you wanna go real fancy, you can use a nested feedback with OPA1652 as frontent, and a beefy fast opamp as the output driver.