Raga Bageshri:

Here is some starting material from Subba Rao's book:

Hindustani
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Belongs to kafi (carnatic melakartha: karaharapriya) taht and uses: Tiwra Ri, 
Komal Ga, Shuddha Ma, Tiwra Dha and Komal Ni. 

There are three types of bAgEshrI according to the swaras used in the 
aarohana, avarohana..

Type 1: 
	Aaroh:   s m g m d n s 
	Avaroh:  s n d m g r s

Type 2:
        Aaroh:   s r g m d n s
        Avaroh:   s n d m g r s

This type is also called shuddha bAgEshrI.

Type 3:
        Aaroh:   s r g m d n s
        Avaroh:  s n d m p d m g r s

Pa is used in a vakra fashion.
Some use tivra ni when halting on tara shadja. This sounds nice 
and no harm is done to the rAgabhAvA.

In all types,

vAdi:    Ma
samvAdi: Sa

The phrases "m d" and "m n" add beauty to the raga. 

rAga Abhogi / Abhogi kanada ( came from carnatic abhogi) also resembles 
bageshree. In abhogi, ni is dropped completely. 

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* Subbarao also mentions that bahAr and carnatic rItigowlA are close to
  some versions of bageshree.

* Avarohanam of this rAgA is not well captured in the above three versions. 
  I have heard phrases like "m p d g", "m d n d p g" which bring the beauty
  of this rAgA out.

* Subba Rao's book does not talk about the carnatic version of bAgEshrI 
  The popular carnatic aarohanam and avarohanam are:
  s g m d n s  ::  s n d m p d g m r s

* Yes, there are several versions of bAgEshrI in different gharAnAs. Type 2
  for instance has exactly the same set of aarohanam and avarohanam as 
  carnatic shrIranjani. There is a strong bAgEshrE flavour even in this
  version, which makes is sound very different from shrIranjani.