There's several contexts where a /ə/ might appear at the end of a word:
1. Wherever an historic /ə/ was present
Unlike English that added ⟨e⟩ at the end of some words to indicate the previous vowel was long (O.E. mus, mys to modern mouse, mice), a graphical ⟨e⟩ in French always correlates with a vowel that used to be pronounced.
Because the loss of word-final /ə/ was relatively recent, and didn't affect all dialects (nor all dialects in the same way), you can still find many modern speakers that will pronounce those /ə/, either all or most of the time (this is particularly frequent in the Southern half of France) or sporadically, especially in elevated registers or when reading aloud.
In songs and poetry, those /ə/ can get pronounced or skipped as fits the metric, and for older compositions because they date from a period where they'd almost always been pronounced. For example, there's a nursery rhyme that's still taught in kindergarten that starts by repeating "Frère Jacques" in 4 syllables, even here in Belgium when those vowels would be systematically skipped in normal speech. You're right that this would be more common in the past, but songs can break the rules.
There's also a few words, usually not used at the end of sentences, whose final /ə/ has stabilised, and is now often or always pronounced at the end of a sentence: the object pronoun -le in imperatives, ce in locutions like "sur ce" and "et ce", and prepositions and particles like "est-ce que", "parce que", "jusque" or "entre".
2. To break up consonant clusters
In careful speech, word-final consonant clusters are reinforced by a /ə/ when followed by a consonant: double consonne, "piste cyclable". In casual speech, the cluster often get simplified instead (doub' consonne, pis' cyclable). In France French, this can extend to words that didn't etymologically end with a vowel like "ours blanc" prounonced "ours[ə] blanc".
3. Prepausal schwa
A relatively recent tendency in the central dialects of France French is to insert a vowel (this can be [ə], but also [œ, ø, ɐ̃]) at the end of sentences or phrases, especially those that end in a consonant. This is probably responsible for the majority of apparent ⟨e⟩ you hear pronounced (although it has no relationship to the orthography of the word and can potentiallly appear after arrives, paris, taf, mal and loger in the following sentence: "Quant t'arrives à Paris sans taf, t'as du mal à te loger")