Without getting into the theory, these are some generalizations without knowing anything about the room, the speaker type, the number of drivers, the driver's parameters or the listening postition:
Use the Bessel setting for a better transient response. If you like the snap of a snare drum or the pluck of guitar strings to be emphasized (poor word, just as a reference), then a better transient response may be obtained with the Bessel. If the drivers exhibit poor behaviour (distortion) at frequency extremes, this setting may not be the best choice.
Use the Butterworth if you want to increase the size of the off-axis response (sweet spot); or if you have a large room and need to tame the lower frequency room modes.
Use the L-R if you want to give a slight boost to the bass response of a speaker or if you are using a turntable and need to minimize low or subsonic frequencies such as rumble and flutter. If you find that the Butterworth sounds good in your setup, then the L-R will probably slightly improve on it without any trade-offs and may be the better choice. But the L-R may not work as well as the Butterworth in a larger room.
Again, these are just generalizations, YMMV.
Use the Bessel setting for a better transient response. If you like the snap of a snare drum or the pluck of guitar strings to be emphasized (poor word, just as a reference), then a better transient response may be obtained with the Bessel. If the drivers exhibit poor behaviour (distortion) at frequency extremes, this setting may not be the best choice.
Use the Butterworth if you want to increase the size of the off-axis response (sweet spot); or if you have a large room and need to tame the lower frequency room modes.
Use the L-R if you want to give a slight boost to the bass response of a speaker or if you are using a turntable and need to minimize low or subsonic frequencies such as rumble and flutter. If you find that the Butterworth sounds good in your setup, then the L-R will probably slightly improve on it without any trade-offs and may be the better choice. But the L-R may not work as well as the Butterworth in a larger room.
Again, these are just generalizations, YMMV.