Russian Spelling

As we have seen, palatalization of consonants (hard v. soft) is phonemically meaningful in Russian, that is, whether a consonant is hard or soft can be the only feature distinguishing two different words. But because the orthography (writing system) uses the same consonant for both (rather than 14 extra letters), they use a diacritic - the soft sign or hard sign Ь Ъ - to mark it. Once the Ъ was written after every hard consonant not followed by a vowel, but now it is only written before the so-called "soft vowels" after prefixes ending in hard consonants. That is, now Russian orthography assumes a consonant to be hard unless it is explicitly marked as soft.

The five Russian vowels have two letters each, one of which says "the consonant before me is hard" and one which says "the consonant before me is soft ... or Y (Й)." It's that "or" that is problematic and makes the Ъ not obsolete; it's needed to show the combination of hard consonant + Й.Without it, you would think that подъехать had a soft Д (as if подехать)instead of a hard one followed by a Й (подйэхать would be the purely phonetic spelling). And съесть to eat would sound exactly like сесть to sit down.

At any rate, the problem for Russian spelling arises because five of the consonants are not paired for softness, and three - к г х (the velars) - never use the soft sign. The unpaired five are the sibilant (husher) consonants. Of those, ш ж ц are always pronounced hard; щ ч are always pronounced soft. But the choice of which vowel to write has nothing to do with that, and is ruled instead by historical phonology. Fortunately, it can be condensed into some invariable rules:

Spelling Rule 1: Ы - И

Never write the letter "Ы" after the letters 'Г, К, Х, Ж, Ч, Ш, Щ' instead use "И"

Spelling Rule 2: unstressed O - E

Never write an unstressed "O" after the letters 'Ж, Ч, Ш, Щ, Ц' instead use "E"

Spelling Rule 3: Я - А

Never write the letter "Я" after the letters 'Г, К, Х, Ж, Ч, Ш, Щ, Ц' instead use "А"

Spelling Rule 4: Ю - У

Never write the letter "Ю" after the letters 'Г, К, Х, Ж, Ч, Ш, Щ, Ц' instead use "У"

Notes:
Number 2 is the most complicated rule, because you have to pay attention to stress: в большом хорошем доме "in a big nice house".
Generally the letter Ë will be used after Ж to show stressed O. (It used to be a soft consonant.)
Remember that Ц can be followed by Ы. 

Here is a page from Alpha Dictionary: Russian Spelling Rules.
Here is one from Russian Lessons.net: Spelling Rules.