This is not about the phonetics of nasalised vowels, but rather about the wide variety of devices for representing them in different orthographic systems.
Let’s start with the superscript tilde, as seen in the picture to the left. This is, of course, the IPA approved diacritic for nasalisation, but not very many languages use this in their orthography. Portuguese is probably the most widely known example of a language that does, having <ã> and <õ>. Guaraní, an Amerind language spoken mainly in Paraguay, uses the tilde for nasalised vowels <ã ẽ ĩ õ ũ ỹ>, the last of these representing [ɨ̃]. In fact, the devisers of the orthography for this language must have been inordinately fond of the tilde. They used it not only for <ñ>, which as in many other languages represents [ɲ], but also for <g̃>, which represents [ŋ]. I don’t know of any other languages which use this combination.
Another fairly common way of representing nasalised vowels is the ogonek, as in Polish <ą> and <ę>, usually with the phonetic values [ɔɰ̃] and [ɛɰ̃] respectively. The closely related language Kashubian has the first of these, but also <ã>, representing [ã] or [ɛ̃].
Quite a few North American languages use ogonek for vowel nasalisation. Apache, spoken in Arizona and New Mexico is a good example, having <ą ę į ǫ ų>. Other North American languages, such as Lakota and Dakota have an alternative strategy, the first using a following <ŋ> and the second a following <ƞ>. This “empty letter” strategy is also used by Breton, which indicates vowel nasalisation with a following <ñ>
I have come across only one language (so far) which uses a diaeresis (aka trema, aka umlaut) for vowel nasalisation. This is Onöndowága (also known as Seneca) spoken by a small number of people in New York State and Canada. This has <ë ö>, and, rather confusingly, <ä>, which represents the oral vowel [æ]. This language is one of only two that I have found which uses <s̈> (to represent [ʃ]).
A few languages use underlining to indicate nasality. Chickasaw, spoken mainly in Oklahoma, uses <a i o u> and Choctaw (Oklahoma, Mississippi, Louisiana and Tennessee) uses <a i o ʊ>.
I am sure I have missed a couple of other methods of indicating vowel nasality. Please do let me know of any I haven’t covered.