Butterworth - Some of the characteristics (Pro and Con) are listed below.

• Monotonic amplitude response in both passband and stopband.
• Quick roll-off around the cutoff frequency, which improves with increasing order.
• Considerable overshoot and ringing in step response, which worsens with increasing order.
• Slightly non-linear phase response.

Linkwitz-Riley

• Absolutely flat amplitude response throughout the passband with a steep 24 dB/octave rolloff rate after the crossover point.
• The acoustic sum of the two driver responses is unity at crossover. (Amplitude response of each is -6 dB at crossover, i.e., there is no peaking in the summed acoustic output.)
• Zero phase difference between drivers at crossover. (Lobing error equals zero, i.e., no tilt to the polar radiation pattern.) In addition, the phase difference of zero degrees through crossover places the lobe of the summed acoustic output on axis at all frequencies.
• The low pass and high pass outputs are everywhere in phase. (This guarantees symmetry of the polar response about the crossover point.)
• All drivers are always wired the same (in phase).
• Not "Linear Phase", meaning that, the amount of phase shift is a function of frequency. Or, put into time domain terms, the amount of time delay through the filter is not constant for all frequencies, which means that some frequencies are delayed more than others.

Bessel - A Bessel filter is a type of analog linear filter with a maximally flat group/phase delay (maximally linear phase response), which preserves the wave shape of filtered signals in the passband. This make the Bessel filter well suited for audio crossover networks.

Solen Split - Theoretically, the Butterworth will have more overlap around crossover frequency, so there will be "bump" at 4kHz, may be 3dB. OTOH, the other one has less bump, could be 0dB or -3dB at 4kHz.
As 4kHz is usually the location of breakups, and our ears are very sensitive to this frequency, and the filter is too simple, it is better to lower the response around this frequency, so I will prefer the "Solen Split", or even much lesser bump.

Legendre

Chebyshev - having a steeper roll-off and more passband ripple (type I) or stopband ripple (type II) than Butterworth filters. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter,[citation needed] but with ripples in the passband.

Gaussian - a Gaussian filter is a filter whose impulse response is a Gaussian function (or an approximation to it). Gaussian filters have the properties of having no overshoot to a step function input while minimizing the rise and fall time. This behavior is closely connected to the fact that the Gaussian filter has the minimum possible group delay. It is considered the ideal time domain filter, just as the sinc is the ideal frequency domain filter.[1] These properties are important in areas such as oscilloscopes[2] and digital telecommunication systems.

Linear Phase - Linear phase is a property of a filter, where the phase response of the filter is a linear function of frequency. The result is that all frequency components of the input signal are shifted in time (usually delayed) by the same constant amount, which is referred to as the phase delay. And consequently, there is no phase distortion due to the time delay of frequencies relative to one another.