LDO Output Capacitor Selection and ESR Consideration
What frequency is this part referring too for it's ESR ratings?  


and 


  1) What frequencies are the datasheets referring to when they rate the
  ESR value? Regulator 1 and 2 seem to have different opinions. AVX Corp
  datasheet lists all tantalum caps ESR at 100kHz. Is this the industry
  standard frequency for LDO ESR ratings too?


The problem with 'low ESR' is its a nebulous marketing term, if your really interested in finding what the ESR is for the capacitor, the frequency and impedance for the capacitor must be known. Usually that means actually digging through the capacitors datasheet or requesting manufacturer impedance information (by request I mead looking for the information on their website)

Usually they mean 10kHz to 100kHz but this can vary from LDO to LDO there is no industry standard. Its from the voltage control loop and it varies from design to design.

So from a design perspective this means digging through the datasheet and finding the ripple and the ESR of the capacitor that they used and ensuring that the ESR is lower than that value. So I would find the lowest ESR capacitor that is available on the market and throw that in your design. How do you do that? compare some capacitors and find the one that has the best impedance curve or a lower impedance curve than the capacitors recommended. 


From ESR stability and the LDO regulator

A capacitor ESR graph looks like the one below, at some manufacturers you can compare these values. or find them in datasheets to get a feel for ESR.




  2) I have read app notes that say not to use 0.1uF bypass caps on the
  output because you can end up with the pole and zero cancelling each
  other out and the regulator could end up unstable. But most schematics
  I see have one. Is this because the designer calculates what the zero
  and pole is or is this normally done in error?


Remember that instability is only a problem if you have transient loads and at the frequency range of interest. Instead of worrying so much about ESR, you should be worried about ripple and under what conditions this can happen. If you do have transient loads (especially transient from 1kHz to 1Mhz see above plot) and you are concerned about the ripple then you do need to worry about the ESR. If ripple isn't a concern then don't worry about it. 

The manufacture is trying to guarantee their ripple specs, they don't want engineers telling them that their products are defective because there is too much ripple at higher frequencies. 

If you really want to know where the pole comes from:


  The pole is at higher frequencies (as shown in the plot above, which
  applies to most regulators but not all). Stated another way, the
  system must have sufficient phase margin, i.e., the amount of phase
  shift remaining until 360° degree when the gain is at 0 dB. Since each
  pole contributes 90° of  phase shift and 20dB/decade (or –1) rolloff
  in gain, a three-pole, high gain system requires compensation in order
  to be stable. A regulator is  nconditionally stable (i.e., has
  sufficient phase margin) if the open loop gain curve rolls off at
  20dB/decade (i.e., like a single polesystem) before crosses 0 dB. The
  most common method of compensation is to insert a zero inthe system to
  cancel the phase shift and rolloff of one of the poles. Since an LDO
  already requires an output capacitor for normal operation, using the
  output capacitor’s ESR is typically the simplest and least expensive
  method for generating this zero.


Also From ESR stability and the LDO regulator  


  3) Since Regulator #2 says not to use high ESR caps, does that mean
  regulator #1 and #2 can't use the same output cap? Since #1 would
  require a minimum ESR greater than the maximum ESR of #2? Even though
  2 doesn't specify ESR range.


It might, if they have different types of control loops then you might have a problem specifying the same capacitor.  Make life simpler for yourself, sample some LDO's throw them on the board in question (with the same capacitor) and measure the ripple under operation. Use a capacitor with a reasonably low ESR