Strange that you mention GIF but don't call it evil. Reducing a general image to GIF format is usually far more lossy

I agree with this, for general-purpose images. I would never, for example, use GIF for icons or wallpaper or photographs.

Furthermore, GIF is evil due to its licensing.

If the licensing were enforced in anything similar to the fashion that was briefly threatened, I would agree. As it stands, in practice, the licensing issues around GIF are a minor caveat, nothing like the horrific issue that is JPEG compression.

PNG is a replacement for GIF - not for JPEG.

No, PNG is a replacement for both GIF and JPEG, as well as TIFF, BMP, XPM, PCX, and so on and so forth. The only other image formats we still need besides PNG are more advanced formats (that support things like layers (XCF) or color-selective opacity, vector formats (POV, SVG, and so forth), and of course the venerable text/plain format for ASCII graphics, which gets better compression than all the others.

JPEG itself isn't evil.

The use of JPEG lossy compression is evil. Lossless JPEGs aren't evil, but with wide PNG support these days (wide if you don't need the alpha channel, which JPEG doesn't have anyway), they're rather pointless.

The lossy compression is usually not something you notice

Notice? NOTICE? It's not a question of noticing. It's a question of being able to suppress my negative reaction long enough to look at the image. When an image (any image, except maybe a solid block of exactly one colour, though of course it's worst for text) is compressed lossily, the compression distortion stands out as *the* predominant feature of the image. It is the first thing I notice. It stands out and calls attention to itself and cannot be ignored.

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