Does it vary based on the street person and their condition.   As some street people actually have a place to live ranging from a home, to a room, or an abandon building in which they are squatters while other live under the highway or in wooded areas.  

How they live and what brought them to their present state impacts what they find valuable.
In the case of squatters or street people who have a traditional form of shelter from renting a room to splitting an apartment or staying in someone's home we have seen possessions range from bikes to cell phones  and basic furniture to vcr's.  It is the more extreme cases that are more interesting.  

What is a valuable possession to someone that lives under a highway overpass?  What are you willing to worry about when you carry your possessions around in a shopping cart?  Are they things that you need to survive and make your life easier, memories from a prior portion of your life, or things that have value to other street people so you can trade.

I went out and found a bum shopping cart and  inventoried the possessions.   Here is a partial list of what I found tied to a telephone pole to ensure someone did not steal the valued possession of this street person.  

It is not a complete list as I was willing to reach in and move things around so it is visible to the naked eye.

Winter Hat & Coat
Sleeping bag wrapped in garbage bags
Blanket
Gloves
Home cleaning supplies including
     Oven cleaner
     Pledge furniture polish
     An off brand of metal cleaner

The head of a mop, the fluffy white part not the pole.
Cassette tapes
fitted bed sheet
day planner
empty bottles of sprite, water, coke and a sports drink
a really dirty and stained pillow
long sheets of plastic
a tarp
photos
old magazines
a make up bag
personal care products: face scrub, compact, face creams, body lotion, deodorant and other about 12 different bottles of things related to every orifice and surface of the human body for both genders.
Cd case
A plastic fly swatter
Winter jacket
Track suit
Wind breaker
Tennis ball
Extension cord
Rubber gloves
A scrub brush
Heavy metal pipe
Shoe boxes, shoe boxes, shoe boxes all stuffed with things and then wrapped in plastic