It looks like it should be "h". Check the spelling here.

If you google

Шәһәр filetype:pdf

you can see it in actual use with various fonts.

yes I have already seen it use with various fonts but it is not obvious from that survey wether the glyph ought to be tall (like h) or small.

And to your list of small versions, you can add Monaco, Consolas and Calibri. Ok, that't a bit puzzling, I agree.

The fonts ParaType developed for the "Public Fonts of the Russian Federation" project both have the ascender, and they should know, right? It being especially made for Russian minority languages and all.

http://www.paratype.com/public/

I have looked at a few of those pdf documents found by google, some being official documents, some being newspapers, and all use a font with the tall version for the lowercase.

Partly to differentiate it from the Latin "h",
what about giving it an ascender but making
the body short anyway? :-)

hhp

I'm familiar with this letter having an ascender, and only the Apple fonts seem to have the short form. I couldn't find the original Unicode proposal document for this character -- possibly it was included in one of the large early batches of pre-encoded characters --, but did find the proposal document for the related shha with spike descender, and the samples used to illustrate the proposal all have an ascender.

I also regarded the ascender form canonical, until I saw text on the Tatar Wikipedia site:

damn –

N O W I’M G E T T I N G REALLY A N G R Y A B O U T T H I S S I T E’ S I M A G E U P L O A D B E H A V I O U R

(look here: http://tt.wikipedia.org/wiki/ – is it Arial?)

Here is what I see in Safari

[Added] The page is http://tt.wikipedia.org/wiki/Шәһәр.

You can see them all in the character viewer (at least on 10.5 and 10.6)

Thank you Michel, I was about to show something similar from Firefox.

Typographically, I still tend to prefer the ascending form. The sound is actually the same like that of ‹h› (in German, in English).
But there must be a reason for the confusion that half the faces have it small and the other half having it tall. – ?

Here is a relevant message written in 2003 by Peter Kirk: http://unicode.org/mail-arch/unicode-ml/y2003-m12/0405.html
I have no idea why all those shapes, there is even one that looks like Hrant's suggestion.

I ran a quick check on what's installed here, and it seemed to be mostly Microsoft and Ascender faces that have the flat 'h'.

Andreas, I would propose repeating the "h" shape for this glyph, and adding the other variant as an alternate.

The low glyph is available as an alternate in Charis SIL and is called uni04BB.UCStyle.