I don’t think it is possible to write rules that would allow you to predict when, in general, silent letters are going to show up in English words. But it is possible to describe some specific groups of silent letters, and even though the groups sometimes overlap, the groupings still might make things a little more tidy:
First, there is the silent final <e> that is serving some diacritic function: (i) marking long vowels, as in pale, dune; (ii) marking a soft <c> or <g>, as in ounce and gouge; (iii) marking voiced <th>, as in breathe, loathe; (iv) insulating otherwise word-final <u>, <v>, <s>, and <z>, as in tongue, carve, false, gauze; or (v) providing a third letter in words that would otherwise be only two letters long, in violation of the Short Word Rule, as in dye, toe. These diacritical silent final <e>’s are very common: 1,127 of them occur in a database of 6,100 common English words.
Second, there are many and diverse non-diacritical silent <e>’s. Some of these are fossils that have lost an earlier function, often marking feminine gender in French as in finesse, madame, nocturne;. Some are native English fossils: are, gone, none, some. Some are silent final <e>’s following syllabic <l>’s, as in cattle, cradle, kettle, little. The database of 6,100 words contains 221 of these non-diacritical silent <e>’s.
Third, there are several different cases where letters have fallen silent due to the simplification of earlier pronunciations: the <p> in psychology, psychiatry; the <d> in grandfather, handkerchief; the <b> in tomb, womb; the <h> in exhaust, annihilate; the <th> in isthmus; the <t> in castle, whistle.
Fourth, very similar to the preceding, there are some letters that fell silent as the syllables they were in were destressed or eased until they disappeared: the first <o> in the American pronunciation of laboratory, the second <e> in every, the <e>’s in past tense and past participle verbs like fixed, kissed.
Fifth, there are a few words in which silent letters were introduced to bring their spelling closer to their actual or assumed classical spellings: debt, indict, receipt, ptarmigan.
The description above can be supplemented with answers to some of the previous answers about silent letters. I’m afraid that that is about as ruly as I can make it.