The following classification comes from the 19th-century writer Austin Phelps:-

1.  Mode of delivery.  They may be delivered from memory, from manuscript, or extemporaneously.

2.  Occasions on which sermons are delivered.  We may speak of ‘ordinary’ and ‘occasional’ sermons.

3.  Subjects.  Sermons may be doctrinal, practical, historical, and philosophical.  But these distinctions can be mischievous.

4.  Audience.

5.  Faculties of mind.  Sermons may be supposed to be addressed to the imagination, the passions, the will, and so on.

6.  Use of texts.  Sermons may be topical, textual, expository, and inferential.

‘The topical sermon is one in which a subject is deduced from the text, but discussed independently of the text.  The textual sermon is one in which the text is the theme, and the parts of the text are the divisions of the discourse, and are used as a line of suggestion.  An expository sermon is one in which the text is the theme, and the discussion is an explanation of the text.   The inferential sermon is one in which the text is the theme, and the discussion is a serious in inferences directly from the text: the text is the premise, a series of inferences in the conclusion.’

For example, a topical sermon on Phil 2:12f might take as its theme, ‘The Sovereignty of God in the Work of Salvation’; and discourse on this theme without any further reference to the text.

A textual sermon on the same passage might cover (a) the duty enjoined in the text: work out your salvation’; (b) the individual responsibility for the soul’s salvation implied in the text, ‘work out your own salvation’; (c) the spirit with which salvation should be sought, ‘with fear and trembling’; (d) the dependence of effort to be saved upon the power of God, ‘it is God which worketh in you’; (e) dependence upon God for salvation is the great encouragement to effort for salvation, ‘work, for it is God which worketh in you.’

An expository sermon might explain the text, by inquiring, (a) In what sense is a sinner commanded to achive his own salvation? (b) What is the spirit of fear and trembling in the work of salvation?  (c) In what sense does the text affirm God to be the author of salvation?  (d) What connection does the text affirm between the earnestness of the sinner and the agency of God?

An inferential sermon might assert (a) that salvation is a pressing necessity to every man; (b) that every man is responsible for his own salvation; (c) that every man who is saved does in fact achieve his own salvation; (d) that dependence upon God is a help, not a hindrance, to salvation; (e) the guilt of trifling with religious convictions; (f) the unreasonableness of waiting in impenitence for the interposition of God; (g) the uselessness of lukewarm exertions to secure salvation; (h) the certainty that every man who is in earnest to be saved will be saved.

7.  Mode of treatment of the subject.  A sermon may be

1. 1. 1. explanatory, in which the chief object is to explain a text, say, or a doctrine or duty;
      2. illustrative, in which the chief object is to ‘intensify the vividness of truth’.  In this kind of sermon, pictorial imagination is pre-eminent;
      3. argumentative, in which the chief object is proof, as with many kinds of doctrinal sermons, and also some ethical sermons;
      4. persuasive, in which the key-note is urgency to present action.

This is Phelps’ preferred method of classification.

Phelps, Theory of Preaching (1881), Lecture III.