The texture mapping for these model types is incorrect:
glass panes: See stained_pane_shaded.png and normal_pane_shaded
iron bars: The 2*16px vertical texture on the iron bar when it is not connected is different than when it is connected; happens facing all 4 directions, but the texture change is different
all of these can take from the default generated UV mappings.
Linked issues
is duplicated by 4
Attachments
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Confirmed in 1.8.1-pre5. Cobble & mossy walls also have incorrect mapping (including rotated tops) when connected on 2 or 3 sides.
Another thing which may come under this report, or may not, is that the largest part of tripwire hooks (the bit that attaches to the block it's placed on) has culling enabled on the north face when it should be the south face. This causes darkening issues. I was able to successfully fix the issue myself with a custom model file which removes cullface:north from the north face of the main body, and adds cullface:south on the south face.
The reason I posted it here is because, although it's not really texture mapping, I feel this report would encompass all parts of the same issue if it was 'model errors' instead of 'incorrect texture mapping'. If you try to argue 'one issue per report' against my comment, then please do the same for the title of this report.

@@unknown
Confirmed in 1.8.1-pre5.
BTW: 1.8.1 is released since a day...
Ok I found it in the Mojang Twitter feed since, they hadn't announced it on the main bit of the site. Still confirmed for 1.8.1.
Possibly also part of the same issue: the 2 diagonal brewing stand arms are stretched out long. Caused by an unneeded 'rescale' value in the rotation of those elements. Set to false or remove entirely to fix. Screenshot attached.
@Conor McS This report started as just about the pane/bar issues, but I added more to it as it all goes under the general heading of incorrect uvs in the model files. Brewing stand and tripwire have already been reported, but they're definitely related.
Can confirm for 1.8.7, this issue goes far beyond mapping on the cauldron, though, the model is just generally poorly made.
Faces that can never be visible aren't removed.
Faces that are still visible have culling applied.
UV coords are unneeded and are mapped incorrectly in a way that mimics MC-66485 (removing the UV coords altogether results in no change)
Still confirmed in 15w32c for all mentioned models, including tripwire hook shading with smooth lighting off.
Also a new bug with panes (due to completely reworked but still broken-as-hell models): the bottom of the centre piece is shaded - glass panes are supposed to have shading disabled, and they do for everything except the bottom centre square.
Why is this not even being looked at?
PS: (To Mojang) I have already done a lot of model-fixing for my resource pack if you want an existing source for correct models.
Edit: Attached screenshot
Confirmed for all mentioned blocks, even the new 1.9 pane/bar models, in 15w39c. As mentioned in my previous comment, the bottom face of the centre rod/intersection of every pane is shaded, while the rest of the model is not. I will add a screenshot. Note: this applies to normal panes and all types of stained pane, but not bars.
Why does nobody even consider this issue to exist anymore? It's still there in 15w40b
15w43c: Don't see anything wrong with iron bars. Currently updating the list with detailed explanations of what I think the problem is. Let me know if I miss something.
edit:
Found the issue with glass panes, just should have looked at the attached screenshots 😃
Regarding cauldrons: These are tracked in MC-58405, which is fixed for the next snapshot. Please add your info there if it's not fixed next week.
edit2:
Rergarding hoppers: No description or screenshots provided and I don't see anything obviously wrong with the model. It could be MC-73401. I removed it from the list, anyway.
edit3: Cake is MC-45375.
Fixed cakes & brewing stands.
The glass panes cannot look any different.
Their faces have to be mirrored so the textures of the panes line up. This causes the 'around the corner' texture to not line up in 2 out of 4 directions.
Here's what it's supposed to be.