Four different types of Film openings:

1) Titles on a blank screen:

Historically this was the most common type of title sequence. It usually involves a white type face on a black background. Studios used them because they are low budget and for those who were lazy they were a simple and easy way to include relevant information needed at the beginning of the film.

2) Titles accompanied by still images: 

Although this title sequence is not complex it it however more elaborate than the first type of title sequence. In the past the title borders and images were hand drawn. This title sequence was a way of combining the text with a visual image to hint the genre of the movie. With this type of title sequence studios were more willing to develop elaborate title sequences. The use of music also accompanies the still images in hinting what tone/genre the film is.

3) Titles with moving images: 

The moving image helps to offer either a metaphor or narrative thread. It also introduces the tone or story line to the audience.

4) Titles built around animation and motion graphics:

In the late 1990's the use of animated text in title sequences became popular. With the use of a lot of digital technology and stylized editing the titles were able to become a part of the moving images.
1 - Narrative: The titles are integrated onto the moving image and they begin as the film begins.
2 - Discrete: The titles are integrated onto the moving image but it is still not clear what the story line is.
3 - Stylized: These are highly stylized title sequences.

My favorite type of title sequence:

My favorite type of title sequence is the one built around animation and motion graphics. This is mainly because it is all I have ever known however in general I think this would always be my favorite. The reason for this is because it engages the audience from just the title sequence whilst giving a hint on what genre the film is. I love how the the type face also blends in with the genre and the moving images.