The Running Horse was a popular town centre pub sat comfortably in a side road known as Paragon Street. Sadly, it, and the street disappeared in 1961 with the creation of the bowling alley and multi-storey car park above.
This is the only known photograph of the pub. The swing sign is visible, but that's about the best we're going to get. Unless of course, a better one comes to light. The emblem above the sign looks very much like that of the Phoenix Brewery.

There was also the Fisherman's Arms, which was allegedly at the seafront end of Paragon Street. I am wondering if they were really one and the same.

We had originally thought that Paragon Street was well under the bowling ally car park and shops and all trace removed but further research seems to show that it's still there - and in use, but as an alley to the car park. This may explain its wide width.

According to the Worthing Journal (issue 115) The Running Horse was a popular haunt for generations of actors and actresses appearing in the town.

 Time Line

 A little bit extra

1835 - 1945 Edward Stilwell
1855 - 1878 Frederick George Carter & Harriet
1890 - Henry Finnis
1913 - 1934 Thomas Cornford
1944 - Ernest Cooper
1946 - 1950s - Ernest Whittock Cooper
1950 T Marworthy
1961 - Demolished

1934: Thomas Cornford, landlord of the Running Horse, Paragon Street, and his barmaid, Daisy Conyers D'arcy, were each fined £1 by Worthing Magistrates for serving a drunk. Mr Cornford had been a publican for 41 years - 21 as landlord of the Running Horse.
Worthing Journal issue 94 - 2018